ON THE UNIFORM THICKNESS PROPERTY AND CONTACT GEOMETRIC …pure.au.dk/portal/files/22414043/UTP_Dissertation.pdf · 2010-11-04 · ON THE UNIFORM THICKNESS PROPERTY AND CONTACT GEOMETRIC

ON THE UNIFORM THICKNESS PROPERTY ANDCONTACT GEOMETRIC KNOT THEORY

by

Douglas J. LaFountainApril 14, 2010

A thesis submitted to theFaculty of the Graduate School of

the University at Buffalo, State University of New Yorkin partial fulfillment of the requirements for the

degree of

Doctor of Philosophy

Department of Mathematics

Acknowledgements

I first wish to thank Bill Menasco, who not only directed me toward the initial problemfrom which the current work developed, but whose steady advisement has assisted methroughout. Moreover, his prior investigations of transversal iterated torus knots in thecontact 3-sphere have served very much as a foundation for much of the current work.

Along these same lines, I am indebted to John Etnyre and Ko Honda for their definitionof, and initial investigation of, the uniform thickness property; their work provided the otherpiece of the foundation upon which the current investigation was built. Bulent Tosun alsodeserves acknowledgement for his definition of the lower width of a knot type, which hashelped to crystallize and clarify many of the current results in this thesis. Moreover, bothJohn Etnyre and Bulent Tosun contributed to the proof that positive torus knots indeedfail the uniform thickness property.

I would also like to thank Joan Birman for her encouragement and interest in this work,including her many helpful e-mail communications. I would also like to acknowledge AdamSikora and Xingru Zhang for their helpful roles as members of my dissertation commit-tee. The geometry/topology group, including both faculty and graduate students, in theDepartment of Mathematics at the University at Buffalo in general has been a source ofmathematical and collegial support.

On a more personal note, I wish to thank Jessica LaFountain first for her initial encour-agement toward my pursuit of a doctoral degree, as well as her constant companionship andsupport throughout. I am also thankful for Jim and Kathleen LaFountain, whose academicand personal experiences have been both a source of inspiration for me, as well as guidance.

ii

Contents

Abstract viii

1 Introduction 1

1.1 Historical and mathematical back-story . . . . . . . . . . . . . . . . . . . . 1

1.2 The convex surface-braid foliation dialectic . . . . . . . . . . . . . . . . . . 4

1.3 Preview of subsequent chapters . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Contact geometry and knots 9

2.1 Contact structures on 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Definition of a contact structure . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Total non-integrability and the characteristic foliation . . . . . . . . 9

2.1.3 Tight contact structures . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.4 Universally tight contact structures . . . . . . . . . . . . . . . . . . . 10

2.1.5 Contact structures on R3 . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.6 The standard contact structure on S3 . . . . . . . . . . . . . . . . . 11

2.1.7 Other contact 3-manifolds . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.8 Contact structures supported by open book decompositions . . . . . 14

2.2 Transversal knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Definition of a transversal knot . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 sl: the classical invariant of transversal isotopy . . . . . . . . . . . . 15

2.2.3 The Bennequin inequality . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.4 Transversal knots represented as braids . . . . . . . . . . . . . . . . 15

2.3 Legendrian knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1 Definition of a Legendrian knot . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 tb and r: the classical invariants of Legendrian isotopy . . . . . . . . 18

2.3.3 The Bennequin inequality . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.4 Stabilization of Legendrian knots . . . . . . . . . . . . . . . . . . . . 18

2.3.5 Transversal push-offs of Legendrian knots . . . . . . . . . . . . . . . 19

2.3.6 Legendrian simplicity implies transversal simplicity . . . . . . . . . . 20

2.3.7 Legendrian mountain ranges . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Cablings and iterated torus knots . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Definition of a cabled knot type . . . . . . . . . . . . . . . . . . . . . 22

iii

iv CONTENTS

2.4.2 Two framings on ∂N(K(P,q)) . . . . . . . . . . . . . . . . . . . . . . 222.4.3 Definition of iterated torus knots . . . . . . . . . . . . . . . . . . . . 232.4.4 Iterated torus knots that support ξstd . . . . . . . . . . . . . . . . . 232.4.5 The quantities Ar and Br for an iterated torus knot Kr . . . . . . . 232.4.6 χ(K(P,q)) in terms of χ(K), when P > 0 . . . . . . . . . . . . . . . . 242.4.7 χ(Kr) when Pi > 0 for all i . . . . . . . . . . . . . . . . . . . . . . . 24

3 Convex surface theory and the UTP 273.1 Convex surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Definition of a convex surface . . . . . . . . . . . . . . . . . . . . . . 273.1.2 The dividing set for a convex surface . . . . . . . . . . . . . . . . . . 273.1.3 Existence of closed oriented convex surfaces . . . . . . . . . . . . . . 283.1.4 Existence of convex surfaces with Legendrian boundary . . . . . . . 293.1.5 Giroux Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.6 Standard form for convex tori . . . . . . . . . . . . . . . . . . . . . . 313.1.7 Edge-rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1.8 Bypasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1.9 How bypasses can change dividing sets . . . . . . . . . . . . . . . . . 343.1.10 Isotopy discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.11 How to find bypasses: convex discs and annuli . . . . . . . . . . . . 383.1.12 Relative Euler class . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.13 Classification of tight contact structures for B3, S1 × D2, and T 2 × I 403.1.14 Twist number lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 The uniform thickness property for knots . . . . . . . . . . . . . . . . . . . 433.2.1 Definition of the contact width of a knot . . . . . . . . . . . . . . . . 433.2.2 Definition of the uniform thickness property (UTP) . . . . . . . . . 433.2.3 Ways to fail the UTP . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.4 Definition of the lower width of a knot . . . . . . . . . . . . . . . . . 443.2.5 Results concerning the UTP . . . . . . . . . . . . . . . . . . . . . . . 443.2.6 Results concerning the UTP and simplicity . . . . . . . . . . . . . . 45

4 Results: UTP cabling and classification theorems 474.1 Cabling theorems for the UTP . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.1 Cabling preserves the UTP . . . . . . . . . . . . . . . . . . . . . . . 474.1.2 Cabling preserves Legendrian simplicity and the UTP . . . . . . . . 494.1.3 Cablings less than integral lower width satisfy the UTP . . . . . . . 494.1.4 Definition of a χ-sequence of solid tori . . . . . . . . . . . . . . . . . 514.1.5 Definition of a χ-candidate knot type . . . . . . . . . . . . . . . . . 514.1.6 Positive cabling preserves χ-candidate knots . . . . . . . . . . . . . . 52

4.2 Positive torus knots χ-fail the UTP . . . . . . . . . . . . . . . . . . . . . . . 594.2.1 Definition of χ-failure of the UTP . . . . . . . . . . . . . . . . . . . 594.2.2 Positive torus knots are χ-candidate knots . . . . . . . . . . . . . . . 594.2.3 The χ-sequence exists with non-thickenable terms . . . . . . . . . . 63

CONTENTS v

4.3 UTP classification of iterated torus knots . . . . . . . . . . . . . . . . . . . 65

4.3.1 The χ-sequence Nkr . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.2 Any Kr with Pi > 0 for all i χ-fails the UTP . . . . . . . . . . . . . 65

5 Results: Simple and non-simple iterated torus knots 69

5.1 Transversally non-simple iterated torus knots . . . . . . . . . . . . . . . . . 69

5.2 Legendrian classification of the ((2, 3), (1, 2)) knot . . . . . . . . . . . . . . . 73

5.3 Legendrian simple iterated torus knots . . . . . . . . . . . . . . . . . . . . . 75

5.3.1 Iterated cablings of negative torus knots are simple . . . . . . . . . . 75

5.3.2 Iterated cablings greater than w(Ki) are simple . . . . . . . . . . . . 75

5.3.3 Cablings less than lw(Ki), after repeated cablings greater than w(Ki),are simple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6 Braid foliations and interlocking solid tori 77

6.1 Braid foliations on embedded surfaces . . . . . . . . . . . . . . . . . . . . . 77

6.1.1 The braid fibration of S3 . . . . . . . . . . . . . . . . . . . . . . . . 77

6.1.2 Vertices and singularities . . . . . . . . . . . . . . . . . . . . . . . . 78

6.1.3 Normalization of surfaces in the braid fibration . . . . . . . . . . . . 78

6.1.4 Three types of non-singular leaves . . . . . . . . . . . . . . . . . . . 78

6.1.5 Five types of singular leaves . . . . . . . . . . . . . . . . . . . . . . . 79

6.1.6 Five types of tiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.1.7 Parities of vertices and singularities . . . . . . . . . . . . . . . . . . 81

6.1.8 The graphs G++, G−−, G−+, and G+− . . . . . . . . . . . . . . . . . 82

6.1.9 The graphs Gǫ,δ for closed surfaces . . . . . . . . . . . . . . . . . . . 82

6.1.10 The braid foliation on a tiling is the characteristic foliation Sξ . . . 84

6.1.11 Closed normalized surfaces with tilings are convex . . . . . . . . . . 84

6.1.12 Braid foliations on tori . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.1.13 Braid moves on foliated surfaces . . . . . . . . . . . . . . . . . . . . 84

6.1.14 Manipulation of the braid foliation . . . . . . . . . . . . . . . . . . . 85

6.1.15 Non-standard changes of foliation result from bypasses . . . . . . . . 90

6.2 Interlocking solid tori: An obstruction to simplicity . . . . . . . . . . . . . . 90

6.2.1 Exchange reducible knots . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2.2 Exchange reducibility implies transversal simplicity . . . . . . . . . . 90

6.2.3 Exchange reducibility in the class of iterated torus knots . . . . . . . 91

6.2.4 Homogeneous twisting uniform steps configurations . . . . . . . . . . 92

6.2.5 The G-framing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2.6 Calculating boundary slopes . . . . . . . . . . . . . . . . . . . . . . . 95

6.2.7 Interlocking, homogeneous twisting, uniform steps configurations . . 95

6.2.8 An interlocking solid torus representing the (2, 3) torus knot . . . . 99

6.2.9 Legendrian rectangular diagrams . . . . . . . . . . . . . . . . . . . . 99

vi CONTENTS

7 Results: Interlocking non-thickenables 1037.1 Interlocking non-thickenables for positive torus knots . . . . . . . . . . . . . 103

7.1.1 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.1.2 Boundary slopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.1.3 Standard neighborhoods of Legendrians . . . . . . . . . . . . . . . . 1057.1.4 Cabling annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.1.5 Verification of non-thickening: the second standard neighborhood . . 109

7.2 Cabling theorems for positive interlocking solid tori . . . . . . . . . . . . . . 1107.2.1 Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.2.2 Local positive G-framing cabling operations . . . . . . . . . . . . . . 1157.2.3 Global negative G-framing cabling operations . . . . . . . . . . . . . 124

7.3 Interlocking non-thickenables for iterated torus knots . . . . . . . . . . . . . 1327.4 Legendrian and transversal MTWS for ((2, 3), (1, 2)) . . . . . . . . . . . . . 134

8 Open questions and future directions 1438.1 Positively twisting interlocking solid tori . . . . . . . . . . . . . . . . . . . . 143

8.1.1 General constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.1.2 Sequences of interlocking solid tori . . . . . . . . . . . . . . . . . . . 1438.1.3 Boundary slopes of interlocking solid tori . . . . . . . . . . . . . . . 1448.1.4 Interlocking solid tori as non-thickenables . . . . . . . . . . . . . . . 1448.1.5 Interlocking solid tori for links . . . . . . . . . . . . . . . . . . . . . 144

8.2 The lower width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.2.1 Lower bound for the lower width . . . . . . . . . . . . . . . . . . . . 1458.2.2 Lower width of zero . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.2.3 Integral lower width . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.2.4 Negative cablings and the UTP . . . . . . . . . . . . . . . . . . . . . 145

8.3 The UTP and fibered knots . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.3.1 Strongly quasipositive knots . . . . . . . . . . . . . . . . . . . . . . . 1458.3.2 The figure eight knot . . . . . . . . . . . . . . . . . . . . . . . . . . . 1468.3.3 Open book decompositions of contact 3-manifolds . . . . . . . . . . 146

8.4 Failing the UTP in S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1478.4.1 tb > w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1478.4.2 χ-failing the UTP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1478.4.3 Non-integral contact width . . . . . . . . . . . . . . . . . . . . . . . 1478.4.4 Non-thickenables even though tb = w = lw . . . . . . . . . . . . . . 147

8.5 Legendrian classification of cabled knot types . . . . . . . . . . . . . . . . . 1478.5.1 Classification of iterated torus knots . . . . . . . . . . . . . . . . . . 1478.5.2 Classification of cablings of fibered knots . . . . . . . . . . . . . . . 148

8.6 Other applications of the UTP . . . . . . . . . . . . . . . . . . . . . . . . . 1488.6.1 Hyperbolic knots and the UTP . . . . . . . . . . . . . . . . . . . . . 1488.6.2 Non-transversal global isotopies . . . . . . . . . . . . . . . . . . . . . 148

CONTENTS vii

Abstract

This work is an investigation of the uniform thickness property (UTP) for knots in thecontact 3-sphere.

We establish cabling theorems for the UTP: in particular we show that if a knot typeK satisfies the UTP, then cablings K(P,q) also satisfy the UTP; we also show that if K isa χ-candidate knot type (for potential failure of the UTP), then positive cablings K(P,q)

are also χ-candidate knot types. This leads us to provide a complete UTP classificationfor the class of iterated torus knots, namely showing that an iterated torus knot Kr =((P1, q1), · · · , (Pr, qr)) fails the UTP if and only if Pi > 0 for all i. More specifically, weidentify all non-thickenable solid tori in the class of iterated torus knots which result infailure of the UTP. We then are able to show that failure of the UTP in the class of iteratedtorus knots is a sufficient condition for the existence of transversally non-simple cablings.We also identify large families of Legendrian simple iterated torus knots, including manythat are cablings of knots that fail the UTP.

We then establish cabling theorems for positively twisting, interlocking, steps configu-rations, and use this to show that every non-thickenable solid torus in the class of iteratedtorus knots can be represented by such an interlocking steps configuration. This then al-lows us to establish new knot-type specific Legendrian and transversal Markov Theoremswithout stabilization for the ((2, 3), (1, 2)) iterated torus knot.

viii CONTENTS

Chapter 1

Introduction

The first goal of this introduction will be to provide for the reader a narrative of the ideasand events that have gone into the present work, with a goal of helping the reader tounderstand the mathematical motivation that inspired the current project. The secondgoal of this introduction will be to then give a brief narrative describing the way differentparts of this project have motivated each other. These two narratives will be fairly informal,with the formalities of the mathematics presented in subsequent chapters. I find personallythat I can appreciate mathematics to a greater degree if it is contextualized in some kindof story, and thus I will strive to perform such a contextualization for the reader in thisintroduction.

The third goal of the introduction will be to preview the different forthcoming chaptersso as to prepare the reader for what to expect. Specifically, we will summarize the mainpoint of each chapter, and indicate how it fits into the larger story. I have heavily sub-sectioned each chapter so as to make the work as “skimmable” as the reader so desires, andtherefore the reader is encouraged to regularly reference the table of contents as a guide.

1.1 Historical and mathematical back-story

The present work is in the areas of contact geometry and low-dimensional topology; specif-ically, it is part of a general program of study of contact 3-manifolds using embedded knotsand surfaces. A contact 3-manifold (M, ξ) is a manifold M with a totally non-integrableplane field ξ, called a contact structure, and arises naturally in physics, where knots andsurfaces appear as periodic orbits and solutions to constraint equations. Contact manifoldshave a combination of rigidity and flexibility that allows for enlightening interplay betweengeometry and topology; for example, recently contact geometry was instrumental in Kron-heimer and Mrowka’s resolution of the long-standing property P conjecture [29], and Ghristand Etnyre have used contact geometry to study the topology of fluid flows [16].

The study of knots in contact 3-manifolds has been useful for understanding contact3-manifolds themselves, specifically since Bennequin’s analysis of the unknot establishedthe existence of exotic contact structures on S3 [2]. More recently, Giroux has reinforced

1

2 CHAPTER 1. INTRODUCTION

the importance of contact geometric knot theory by establishing a one-to-one correspon-dence between contact structures on 3-manifolds and open book decompositions of those3-manifolds associated to fibered links [23].

There are two important classes of knots in contact 3-manifolds, namely those knotsthat are tangent to the contact structure (called Legendrian knots), and those that aretransverse to the contact structure (called transversal knots). The current project addressesthe classification of Legendrian and transversal iterated torus knots, which are examplesof fibered knots. This is an area of central importance to contact geometry, as evidencedby Etnyre and Van Horn-Morris’ recent result that contact structures are determined byfibered transversal knots [19]; furthermore, important constructions in contact topology,such as contact surgery, rely on understanding Legendrian knots.

The main problem the present work addresses is the classification of Legendrian (transver-sal) knots up to isotopy through Legendrian (transversal) knots. A topological knot type Khas infinitely many Legendrian (transversal) isotopy classes, and there are classical invari-ants that remain fixed under Legendrian (transversal) isotopy. A knot K whose Legendrian(transversal) isotopy classes are classified by these classical invariants is said to be Legen-drian (transversally) simple. Not all knot types are Legendrian or transversally simple (see[17, 39]); thus an important question is which knot types are simple, and which arenon-simple?

The second problem this work addresses is the following: given a non-simple knot type,and thus two Legendrian (transversal) isotopy classes having the same values for the classicalinvariants, what global isotopies are required to take a representative of one classto a representative of the other? This is an important question, since no classifyinginvariant is yet known that can classify Legendrian and transversal knots in general, yetsuch a classifying invariant would have to be able to detect such global isotopies.

In this present work, the above two questions are brought to bear on cabled knot types ingeneral, and iterated torus knots specifically. To clarify, given a knot K, we can construct anew knot, denoted K(P,q), on the boundary of a neighborhood of K; different integer valuesof (P, q) will yield different knots. We call K(P,q) a (P, q)-cabling of K. Without loss ofgenerality we may assume qi > 0 for all i; Pi may be negative or positive. Iterated torusknots are then defined to be iterated cablings that begin with torus knots (P1, q1); we willdenote a general iterated torus knot by Kr = ((P1, q1), ..., (Pi, qi), ..., (Pr, qr)) to indicatethe specific cabling (Pi, qi) performed at each iteration.

Two very different, but related, technologies are used in the present work to investigatethe Legendrian classification of iterated torus knots and cabled knot types in general. Thefirst techniques used are a combination of those developed by Giroux, Kanda, and Honda,namely using convex surfaces to extract information about the knots embedded in them(see [22, 24, 25]). The second set of techniques are braid foliation techniques developed byBennequin, Birman, and Menasco (see [2, 5, 35]). The first set of techniques were used byEtnyre and Honda in [17] in an initial investigation of Legendrian cabled knot types, whilethe second set of techniques were used by Menasco in [35, 36] in an initial investigation oftransversal iterated torus knots; both of these works have heavily influenced the present

1.1. HISTORICAL AND MATHEMATICAL BACK-STORY 3

work, and we therefore summarize both of these two independent inquiries here.

Menasco studied transversal simplicity in the class of iterated torus knots in [35, 36],and found that the obstruction to transversal simplicity in the class of iterated torus knotswas precisely the existence of particular tori where things got “locked up”, and no furtherdestabilizations or exchange moves could be identified for particular cablings embedded inthose tori. These tori had braid foliations that were standard tilings, but they also satisfieda particular embedding condition, namely they could be constructed as boundaries of solidtori that were neighborhoods of a complex of blocks and discs, arranged so that they wereinterlocking. These interlocking block-disc presentations are such that they prevent certainnon-standard changes of fibration that would lead to further destabilizations of cablings.The example for a solid torus representing the (2, 3) torus knot is what appeared in thatpaper, and in later papers, and is what is shown in Figure 1.1; we will revisit this figure ina moment.

Figure 1.1: Shown is a positively interlocking steps configuration for the (2, 3) torus knot, with stepsize k = 1. The blocks on the far right and far left of the diagram wrap around the back; the blocksat the top and bottom of the diagram are identified. The whole block-disc complex deformationretracts onto a (2, 3) torus knot. This figure is by Menasco.

Meanwhile, Etnyre and Honda had defined the uniform thickness property for knots in[13]; a knot type K satisfies the UTP if every solid torus N representing K can thicken out-ward to a convex torus with a particular foliation induced by the ambient contact structure.They found that the obstruction to Legendrian simplicity in the class of iterated torus knotswas the existence of solid tori that failed to thicken, meaning that slope(Γ∂N ) = slope(Γ∂N ′)for all N ′ ⊃ N . (Here Γ refers to the dividing curves on a convex torus.) Specifically, asolid torus for the (2, 3) torus knot that failed to thicken supported a (2, 3)-cabling thatfailed to destabilize; this (2, 3)-cabling of the (2, 3) torus knot was therefore transversallynon-simple.

Now if one examines Figure 1.1, and tries to thicken the solid torus by pushing right


edges of the gray blocks forward in the braid fibration (or left edges backward) one findsthat one cannot; the interlocking nature of the blocks forces the torus to run into itself,and prevents non-standard changes of fibration, and we are stuck. As a consequence, thegraph G++ which connects positive elliptic and hyperbolic singularities seems to be aninvariant under thickening. So it seems that this interlocking torus is in fact a torus thatfails to thicken in the Etnyre-Honda sense. When we sat down and did the calculations,that is exactly how things worked: both the above torus and the torus on which Etnyreand Honda found the transversally non-simple (2, 3)-cabling had slopes of G++ and Γ bothbeing −2/11 (as measured in a framing where the longitude is not the preferred longitude,but a non-standard longitude coming from the cabling torus). Moreover, the complementof the interlocking solid torus in S3 decomposed into exactly the complement needed to runthe Etnyre-Honda argument that shows that this torus in fact fails to thicken.

This was therefore the beginning point for the current work; specifically, my goal wasto formalize the relationship between the non-thickenable solid tori of Etnyre and Hondaon the one hand, and the interlocking block-disc solid tori of Menasco on the other. Thecurrent work accomplishes this by identifying all non-thickenable solid tori in the class ofiterated torus knots, and showing that each such non-thickenable can be represented byan interlocking block-disc presentation. Along the way, we also prove cabling theorems forboth the UTP and interlocking block-disc presentations that hold outside of the class ofiterated torus knots. Furthermore, we show that failing the UTP in the class of iteratedtorus knots is a sufficient condition for supporting transversally non-simple cablings.

1.2 The convex surface-braid foliation dialectic

Throughout the investigation that has resulted in the present work, there has been a positivefeedback loop between convex surface theory and braid foliation techniques, with bothinforming the other; I will therefore take a moment to describe some key parts of thisdialectic below so as to give the reader an appreciation for how the two technologies haveinformed each other in real-time.

The first question I took up was how to generalize the relationship between non-thickenable solid tori and interlocking block-disc presentations beyond the example in Figure1.1. To this end, Etnyre and Honda in fact found an infinite sequence of solid tori represent-ing the (2, 3) torus knot that fail to thicken; these tori have slope(Γ∂N ) = −(k+1)/(6k+5),where k ∈ N. The above example in Figure 1.1 is where k = 1. I found that I couldconstruct all elements of this sequence of solid tori as interlocking block-disc presentationsby varying the step size of the blocks. To see what this means, in Figure 1.2 is a picture ofthe interlocking block-disc presentation for the (2, 3) torus knot where k = 2 (so here theslope of G++ is −3/17). In that figure, one can see that as we move in succession from onegray block to the next one on top of it, we have to go up 2 blocks (or 2 steps) until the leftedge of the new block is at the right edge of the block where we started. This is what ismeant by step size of 2. One can make the step size equal to k for any k ∈ N, and thusrealize as interlocking block-disc presentations all solid tori that fail to thicken for the (2, 3)

1.2. THE CONVEX SURFACE-BRAID FOLIATION DIALECTIC 5

torus knot.

Figure 1.2: Shown is a positively interlocking steps configuration for the (2, 3) torus knot, withstep size k = 2.

But one can construct such an infinite list of interlocking block-disc presentations forall (P, q) torus knots, and one can calculate that slope(Γ∂N ) = −(k + 1)/(Pqk + P + q) fork ∈ N – in fact, (Pqk +P + q) is simply the number of blocks. For example, in Figure 1.3 isa schematic for the interlocking block-disc presentation of the (3, 4) torus knot with k = 3;so slope(Γ∂N ) = −4/43.

Now Etnyre and Honda had only done the calculations for slopes of solid tori thatfailed to thicken for the (2, 3) torus knot. But because I could construct the solid torithat should fail to thicken for an arbitrary (P, q) torus knot, I knew what to look for whenI tried to generalize their convex surface arguments. My convex surface calculation thatfinds candidates for non-thickenable tori representing (P, q) torus knots is Lemma 4.2.1in Chapter 4, and there the notation Nk refers specifically to a solid torus with slope−(k + 1)/(Pqk + P + q), and with a complement that should prevent thickening; in otherwords, Nk is precisely the interlocking block-disc presentation with step size k.

So I could construct interlocking block-disc presentations for all (P, q) torus knots, anduse calculations in the braid-foliation setting to guide my convex surface calculations. Ithen found that I could construct interlocking block-disc presentations for iterated torusknots, where the block-disc presentation for, say, the ((P1, q1), (P2, q2)) iterated torus knotwas actually contained in the block-disc presentation for the (P1, q1) torus knot. Shown inFigure 1.4 is a picture for the ((2, 3), (3, 2)) iterated torus knot (but note that the second(3, 2) is measured in the non-standard framing). The picture is a little confusing, but notethat the three blocks on the far right that extend vertically to the top of the grid in factcan be flipped forward to lie inside the solid torus representing the (2, 3) torus knot.

From this interlocking block-disc construction, I was then able to calculate that if Kr =((P1, q1), ..., (Pr, qr)) was an iterated torus knot where the Pi > 0, then the solid tori that


Figure 1.3: Shown is a positively interlocking steps configuration for the (3, 4) torus knot, withstep size k = 3.

Figure 1.4: Shown is an interlocking steps configuration representing the (3, 2)-cabling ofthe (2, 3) torus knot.

1.3. PREVIEW OF SUBSEQUENT CHAPTERS 7

should fail to thicken were those that had slopes of the form −(k +1)/(Ark +Br), where Ar

and Br are calculable quantities based on the Pi’s and qi’s (see equation 2.8 in Chapter 1),and k ranged over an infinite subset of positive integers. Then, when I went to construct aconvex surface argument to find solid tori that should fail to thicken, I again knew what tolook for.

So, in summary, it was interlocking block-disc constructions of solid tori coming out ofthe braid foliation machinery that helped me see how to find candidates for solid tori Nk

r

that failed to thicken. Then, knowing what their complements in S3 looked like, I was ableto run convex surface arguments to actually prove that all iterated torus knots Kr withPi > 0 fail the UTP. Moreover, in the standard framing, using a calculation for χ(Kr), onecan show that the slopes of Nk

r actually are positive and equal −(k + 1)/(χ(Kr)) – thisis critical in showing that any iterated torus knot Kr with at least one negative iterationsatisfies the UTP.

1.3 Preview of subsequent chapters

Chapter 2 is an introductory chapter which reviews basic definitions and results that we willneed concerning contact 3-manifolds, Legendrian and transversal knots, as well as cabledknot types and iterated torus knots.

Chapter 3 reviews essential elements of convex surface theory, and contains many keyresults which we will then use in later chapters to prove new results. This chapter alsoreviews the definition of the uniform thickness property (UTP) for knots, as well as keyresults concerning the UTP and cabled knot types due to Etnyre-Honda and Tosun.

Chapter 4 is the first chapter devoted to our new results. In particular, we first provetheorems that show how the UTP, or certain aspects of the UTP, behave under the operationof cabling. This then leads to a complete UTP classification of iterated torus knots; i.e.,we determine precisely which iterated torus knots fail the UTP, and which satisfy the UTP.The techniques used to establish the results in this chapter are solely in the area of convexsurface theory.

Chapter 5 is the second chapter devoted to our new results. In particular, we use resultsfrom Chapter 4 to show that failure of the UTP in the class of iterated torus knots is asufficient condition for the existence of cablings that are transversally non-simple. We alsoprovide a complete Legendrian classification for one of these new transversally non-simpleiterated torus knots, specifically the ((2, 3), (1, 2)) iterated torus knot. We then also showthat there are large families of Legendrian simple iterated torus knots. Again, the techniquesused to establish the results in this chapter are solely in the area of convex surface theory.

Chapter 6 returns to background material; specifically, we review essential elementsof braid foliation techniques and interlocking solid tori. In particular, we show how byusing braid foliations and interlocking solid tori we can identify both transversal and non-transversal isotopies for transversal cabled knot types.

Chapter 7 is the final chapter devoted to our new results. We first show that all non-thickenable solid tori representing positive torus knots can be represented as interlocking


solid tori. We then establish cabling theorems that show how interlocking solid tori behaveunder the operation of cabling. This then allow us to construct interlocking solid tori thatrepresent every non-thickenable solid torus in the class of iterated torus knots. Finally, thisrealization of these non-thickenable solid tori, along with Legendrian rectangular diagrams,allows us to prove a Legendrian and transversal Markov Theorem without stabilization forthe transversally non-simple ((2, 3), (1, 2)) iterated torus knot investigated in Chapter 5.

Chapter 8 concludes with a discussion of open questions and future directions thatnaturally arise from the considerations of the previous chapters.

Chapter 2

Contact geometry and knots

In this chapter we review relevant background concerning contact 3-manifolds, Legendrianand transversal knots, and iterated torus knots. Our two main references from which muchof this material is drawn are [15, 21].

2.1 Contact structures on 3-manifolds

2.1.1 Definition of a contact structure

Let M be an oriented 3-manifold, possibly with boundary. A contact structure on M is atotally non-integrable 2-plane field ξ, given as the kernel of a 1-form α, where α satisfies anon-integrability condition α ∧ dα 6= 0. For a fixed contact structure ξ there will be manyα’s such that ξ = ker α. A fixed α will be called a contact 1-form. The pair (M, ξ) will bereferred to as a contact 3-manifold. All of our contact structures will be oriented positively,meaning α ∧ dα > 0 when acting on a positively oriented basis for TxM .

Given a 3-manifold M and two contact structures ξ1 and ξ2, a diffeomorphism φ :(M, ξ1) → (M, ξ2) is said to be a contactomorphism if φ∗ξ1 = ξ2; in this case (M, ξ1) and(M, ξ2) are said to be contactomorphic.

A contact isotopy of a contact 3-manifold (M, ξ) is a one parameter family of contacto-morphisms φt : M → M such that φ0 is the identity map.

An isotopy of a contact structure will be a pointwise isotopy of the contact planes in a3-manifold M , so that at each point in the isotopy the plane field is a contact structure.

2.1.2 Total non-integrability and the characteristic foliation

Geometrically, the total non-integrability of the contact 1-form means that if S is an orientedsurface embedded in M , then the tangent planes to S will never exactly coincide with allof the contact planes for points in S. Specifically, there will always be some point x ∈ Ssuch that ξx 6= TxS. Thus, any surface S will have a singular foliation induced by thecontact structure, called the characteristic foliation, and denoted Sξ. Singular points inthe characteristic foliation will be points x ∈ S where ξx = TxS; these singular points will

9

10 CHAPTER 2. CONTACT GEOMETRY AND KNOTS

be positive when the orientations of ξx and TxS agree, and negative otherwise. Away fromthe singular points, the characteristic foliation Sξ is obtained by integrating the line fieldξx ∩ TxS for all x ∈ S non-singular.

2.1.3 Tight contact structures

Let D be an embedded disc in a contact 3-manifold (M, ξ). If the characteristic foliationDξ has a single singular point with ∂D being a limit cycle for Dξ, then D is called anovertwisted disc. Figure 2.1 shows an overtwisted disc.

Figure 2.1: Shown is an overtwisted disc.

If there exists an overtwisted disc D embedded in (M, ξ), then ξ is said to be an over-twisted contact structure. If there does not exist an overtwisted disc D in (M, ξ), then ξis said to be a tight contact structure. We will be mostly interested in working with tightcontact structures. The distinction between tight and overtwisted contact structures wasfirst noticed by Bennequin [2], but formalized by Eliashberg [9].

2.1.4 Universally tight contact structures

Let π : M → M be the universal cover of a tight contact 3-manifold (M, ξ), where ξ = kerα.

Then π∗α is a contact 1-form for M . If ξ = ker π∗α is a tight contact structure, then theoriginal contact structure ξ is said to be universally tight. If there exists a finite coverp : Mf → M of M such that ξf = ker p∗α is an overtwisted contact structure, then theoriginal contact structure ξ is said to be virtually overtwisted. It is not known whether everycontact structure which fails to be universally tight is in fact virtually overtwisted, thoughit is true when π1(M) is residually [25].

We will be mostly interested in working with universally tight contact structures, inparticular for 3-manifolds M that are circle bundles over a closed oriented surface S. Forthis kind of manifold M , if ξ is a contact structure that is everywhere transverse to the S1

2.1. CONTACT STRUCTURES ON 3-MANIFOLDS 11

fibers, then the contact structure ξ is said to be a horizontal contact structure. From thework of Honda, horizontal contact structures are universally tight [25].

2.1.5 Contact structures on R3

The standard contact structure ξstd on R3 is the unique tight (positively oriented) contact

structure on R3, and can be easily visualized. Using a cylindrical coordinate system, if we

let α = dz + r2dθ, then ξstd = ker α is shown in Figure 2.2. Proving that this contactstructure is tight is non-trivial, and is due to Bennequin [2].

Figure 2.2: Shown is the standard contact structure ξstd in R3; it is oriented in the direction of

increasing θ. One translates the z = 0 picture to arbitrary z-value in order to obtain the contactstructure for all of R

3. This figure is by S. Schonenberger.

The standard contact structure on R3 is often given in rectangular coordinates as

ker(dz − ydx); this contact structure, shown in Figure 2.3, is contactomorphic to the oneshown in Figure 2.2.

Darboux’s theorem says that locally any contact structure on a 3-manifold is contac-tomorphic to the standard contact structure on R

3. As a result, contact 3-manifolds haveno local invariants; heuristically, this is one of the main reasons that studying contactstructures on 3-manifolds can be useful in uncovering properties of the underlying globaltopology.

An overtwisted contact structure on R3 is given by ker(cos rdz+r sin rdθ) and is pictured

in Figure 2.4. An overtwisted disc can be visualized by taking a disc with 0 ≤ r ≤ π,0 ≤ θ ≤ 2π, and with z = 0 at r = π and z > 0 for 0 ≤ r < π.

2.1.6 The standard contact structure on S3

Let S3 be the unit 3-sphere in R4. Let i : S3 → R

4 be the inclusion map; then α =i∗(1

2(x1dy1 − y1dx1 + x2dy2 − y2dx2)) is a contact 1-form. The standard contact structure


Figure 2.3: Shown is the standard contact structure for R3, given as ker(dz − ydx); the figure is by

S. Schonenberger.

Figure 2.4: Shown is an overtwisted contact structure on R3; the figure is by S. Schonenberger.

2.1. CONTACT STRUCTURES ON 3-MANIFOLDS 13

on S3 is defined to be ξstd = kerα; ξstd is tight, and if one removes a point from S3 thiscontact structure is the standard one on R

3.

The contact 3-manifold (S3, ξstd) can be visualized nicely in terms of the Hopf fibrationassociated with S3. To see this, recall that S3 can be fibered by S1’s so that a Hopf linkforms the two singular fibers of the fibration, and all of the regular fibers have linkingnumber one with respect to the two components of the Hopf link; see Figure 2.5. If onethen associates to each point x ∈ S3 the unique plane that is perpendicular to the fiberpassing through x, one obtains the contact structure ξstd.

Figure 2.5: Shown is a schematic of the Hopf fibration of S3 by S1’s. The two singular fibers formthe cores of the unknotted tori; regular fibers are shown as curves on the tori peripheral to thesingular fibers. The standard contact structure ξstd is everywhere perpendicular to the Hopf fibers.The figure is by J. Birman and N. Wrinkle.

2.1.7 Other contact 3-manifolds

We will see in the next subsection that every closed oriented 3-manifold supports a contactstructure; for the moment, however, we should make mention of some results concerningthe classification of contact structures. We note the following:

• Overtwisted contact structures on a closed 3-manifold M are in 1-1 correspondencewith homotopy classes of plane fields [9].

• There is a unique tight contact structure on the 3-ball [10].

• Complete classification results for tight contact structures have been obtained for the3-torus [28] and lens spaces [13, 14, 24].

• Complete classification results for tight contact structures have been obtained for solidtori and T 2 × I [24].


This last item will be particularly important for our purposes and will be presentedmore thoroughly in Chapter 2.

2.1.8 Contact structures supported by open book decompositions

An open book decomposition of a closed oriented 3-manifold M consists of a link L, alongwith a fibration π : M\L → S1 such that π−1(θ) is the interior of a surface Σθ where∂Σθ = L for all θ ∈ S1. We will all L the binding, and Σθ the pages of the open bookdecomposition.

A fundamental result of Thurston and Winkelnkemper is that every open book decom-position (L, π) supports a contact structure, ξL, where L is transverse to ξL, and ξL canbe isotoped to be nearly tangent to compact subsets of the pages [43]. Since every closedoriented 3-manifold admits an open book decomposition, this means that any such manifoldM can be thought of as a contact 3-manifold (M, ξ) for some contact structure ξ.

Not only are open book decompositions a way of understanding the existence of contact3-manifolds, but they are also a way of understanding isotopy classes of contact structureson a fixed closed oriented 3-manifold. This is the content of the following fundamental cor-respondence due to Giroux: For an oriented closed 3-manifold M , there is a 1-1 correspon-dence between oriented contact structures ξ up to isotopy, and open book decompositions(L, π) up to positive stabilization [23].

2.2 Transversal knots

2.2.1 Definition of a transversal knot

A transversal knot T in a contact 3-manifold (M, ξ) is an embedded S1 that is alwaystransverse to ξ, meaning TxT ⊕ ξx = TxM for all x ∈ T . In the literature, what we call atransversal knot is sometimes called a transverse knot.

We will be interested in transversal knots up to transversal isotopy; a transversal isotopyof transversal knots is an isotopy φt : S1 → M such that for all t ∈ [0, 1], φt(S

1) isan embedded transversal knot. Two transversal knots T0 and T1 representing the sametopological knot type are said to be transversally isotopic if there is a transversal isotopytaking T0 to T1.

Just as the classification of topological knots up to topological isotopy is equivalentto the classification up to ambient isotopy, so the classification of transversal knots up totransversal isotopy is equivalent to the classification up to ambient contact isotopy. Here wemean T0 and T1 are ambient contact isotopic if there is a one parameter family φt : M → Mof contactomorphisms such that φ0 is the identity map and φ1(T0) = T1. This perspectivewill be helpful at certain key points in what follows.

2.2. TRANSVERSAL KNOTS 15

2.2.2 sl: the classical invariant of transversal isotopy

Given a transversal knot T , its topological knot type is clearly invariant under transversalisotopy. There is also another classical invariant of transversal isotopy, called the self-linking number, and denoted sl. This self-linking number associated to a transversal knot isan integer, and is obtained as follows. Let Σ be a Seifert surface for T ; we thus are assumingthat the knot type is null-homologous. Then since any orientable 2-plane bundle is trivialover Σ, ξ|Σ forms a trivial 2-dimensional bundle, and we can find a non-zero vector field vover Σ in ξ. Let T ′ be a copy of T obtained by taking a push-off of T in the direction of v.The self-linking number sl(T ) is then defined to be the linking of T ′ with T .

Thus for a given topological knot type K, the transversal isotopy class of a transversalknot T determines its self-linking number. Conversely, if, for a given topological knottype K, self-linking numbers determine all transversal isotopy classes, then K is said tobe transversally simple. A knot type K is thus transversally non-simple if there exists twotransversal isotopy classes at the same value of sl.

Transversally simple knots include the unknot [11], as well as torus knots and the figureeight knot [18]; transversally non-simple knots include large classes of 3-braids [6], as well asthe (2,3)-cabling of a (2,3) torus knot [17]. A major part of the present work is to establishnew large classes of both transversally simple and non-simple iterated torus knots.

2.2.3 The Bennequin inequality

Let (M, ξ) be a tight contact 3-manifold, T a (null-homologous) transversal knot embeddedin (M, ξ), Σ a minimal genus Seifert surface for T , and χ(Σ) the Euler characteristic of Σ.Then the following inequality holds, and is called the Bennequin inequality:

sl(T ) ≤ −χ(Σ) (2.1)

Thus in a tight contact 3-manifold, any topological knot type K has a maximal self-linking number, denoted sl(K); this number is a topological invariant for the knot type.

The Bennequin inequality is not always sharp; for example, it is sharp for positive torusknots, but not for negative torus knots [18].

2.2.4 Transversal knots represented as braids

An Alexander theorem for transversal knots

We will be mostly interested in studying transversal knots in (S3, ξstd); we can model thiscontact 3-manifold using Figure 2.2, and in our minds add a point at ∞ so that the z−axisis in fact one of the singular Hopf fibers from Figure 2.5. We will call this singular Hopffiber A := z − axis ∪ {∞}.

With this model in mind, note that associated to A in (S3, ξstd) is a braid fibration ofS3 where A is the braid axis, and the complement of A is a bundle of disc fibers over S1,with all of the other Hopf fibers transverse to these discs. Furthermore, the orientations ofA and the bundle of disc fibers agree using a right-hand rule.


We know that given any topological knot K in S3, any (oriented) representative ofK can be braided with respect to A via a topological isotopy, so that the orientations ofthe representative of K and the disc fibers agree. This is first due to Alexander, and isoften referred to as Alexander’s theorem [1]. There is a similar result for transversal knotsrepresenting a knot type K. Specifically, Bennequin showed that if T is a transversal knot,then there is a transversal isotopy so that after the isotopy, T is braided positively withrespect to the braid fibration described above [2]. This can be thought of as an Alexandertheorem for transversal knots in S3.

We note that a similar Alexander theorem for transversal knots occurs in contact 3-manifolds supported by open book decompositions, whereby a transversal isotopy can posi-tion any transversal knot as a braid with respect to the binding, and accompanying fibration,of the open book decomposition [41].

Calculation of self-linking number

For a transversal knot T represented as a braid, there is an easy way to calculate itsself-linking number. Specifically, if w(T ) is the writhe of the knot (the signed sum of itscrossings, which is equal to the algebraic length), and n(T ) is the braid index (the numberof intersections with each disc fiber), then we have:

sl(T ) = w(T ) − n(T ) (2.2)

A Markov theorem for transversal knots

There are three basic isotopies of braids:

• Braid isotopy: This is isotopy in the complement of the braid axis A, so that at eachpoint in the isotopy the knot remains braided.

• Positive stabilization/destabilization: This either adds (stabilization) or takes away(destabilization) a trivial loop around the braid axis with an accompanying positivecrossing. Figure 2.6 is a template for positive destabilization.

• Negative stabilization/destabilization: This either adds or takes away a trivial looparound the braid axis with a negative crossing.

Note that negative stabilization increases n by one and decreases w by one, and hencedecreases sl by 2; hence it is not a transversal isotopy (nor is negative destabilization, whichincreases sl by 2). Braid isotopy and positive stabilization/destabilization keep sl constant,and in fact can be accomplished by transversal isotopies.

Markov’s theorem for braids says that any two braids representing the same knot typeare related by braid isotopy and stabilization/destabilization [7]. There is a similar resultfor transversal knots represented as braids: Any two transversal knots represented as braidsare related by braid isotopy and positive stabilization/destabilization [40]. This thereforegives a Markov theorem for transversal knots represented as braids.

2.3. LEGENDRIAN KNOTS 17

Figure 2.6: Shown is a template for positive destabilization. Positive stabilization is the reverseoperation; negative stabilization/destabilization is similar, but where the trivial loop involves anegative crossing. This figure is by Birman and Wrinkle.

The exchange move

There is a fourth isotopy that is useful in large part because it does not change the braidindex of a braid, and is in fact a transversal isotopy as well; it is the exchange move, and isillustrated in Figure 2.7.

Figure 2.7: Shown is a template for the exchange move. This figure is by Birman and Wrinkle.

2.3 Legendrian knots

2.3.1 Definition of a Legendrian knot

A Legendrian knot L in a contact 3-manifold (M, ξ) is an embedded S1 that is alwaystangent to ξ, meaning TxL ⊂ ξx for all x ∈ L. All of our Legendrian knots will be oriented.

We will be interested in Legendrian knots up to Legendrian isotopy; a Legendrian isotopyof Legendrian knots is an isotopy φt : S1 → M such that for all t ∈ [0, 1], φt(S

1) isan embedded Legendrian knot. Two Legendrian knots L0 and L1 representing the sametopological knot type are said to be Legendrian isotopic if there is a Legendrian isotopytaking L0 to L1.


Just as for transversal knots, the classification of Legendrian knots up to Legendrianisotopy is equivalent to the classification up to ambient contact isotopy.

2.3.2 tb and r: the classical invariants of Legendrian isotopy

Besides topological knot type, there are two classical invariants of Legendrian isotopy, calledthe Thurston-Bennequin number and rotation number, and denoted tb and r respectively.Both take on integer values, and their definitions are as follows.

The Thurston-Bennequin number measures the twisting of the contact planes around Lwth respect to the Seifert framing. Specifically, let L be a (null-homologous) Legendrianknot, Σ its Seifert surface, and ν the normal bundle of L. If νx is the plane normal to L ateach point x ∈ L, then ξx ∩ νx gives a framing of the trivialization of ν; call this line bundlel. However, there is also a framing coming from a Seifert surface for L; the twisting of lwith respect to this Seifert framing is tb(L).

The rotation number measures the winding number of the knot in a trivialization of ξto L. Specifically, the trivialization of ξ|Σ induces a trivialization of ξ|L = L × R

2. SinceL is oriented, associated with L is a non-zero tangent vector field v. The winding numberof v as it traverses the knot is the rotation number r(L). Note the rotation number forL depends on the orientation of L; switching orientation changes the sign of r. (However,since not all knots are isotopic to their orientation-reverses, reversing orientation may resultin a new knot type.)

If for a given topological knot type K, the ordered pair (r, tb) determines all Legendrianisotopy classes, then K is said to be Legendrian simple. A knot type K is thus Legendriannon-simple if there exists two Legendrian isotopy classes at the same value of (r, tb).

2.3.3 The Bennequin inequality

Let (M, ξ) be a tight contact 3-manifold, L a (null-homologous) Legendrian knot embeddedin (M, ξ), and Σ a minimal genus Seifert surface for L. Then the following inequality holds,and, just as in the case of transversal knots, is called the Bennequin inequality:

tb(L) + |r(L)| ≤ −χ(Σ) (2.3)

Thus in a tight contact 3-manifold, any topological knot type K has a maximal Thurston-Bennequin number, denoted tb(K); this number is a topological invariant for the knot type.

We note that the existence of an overtwisted disc implies the existence of a Legen-drian unknot with tb = 0 and thus violates the Bennequin inequality; thus the Bennequininequality holds if and only if the contact structure is tight.

2.3.4 Stabilization of Legendrian knots

One can decrease the tb of any Legendrian knot L by two well-defined local moves:


• Positive stabilization: This move, denoted by S+, decreases tb by one, but increasesr by one. The actual move involves passing through a local disc with characteristicfoliation as in Figure 2.8.

• Negative stabilization: This move, denoted by S−, again decreases tb by one, butdecreases r by one. The actual move involves passing through a local disc as in Figure2.8, but with the parity of all the singularities reversed.

Figure 2.8: Shown is positive stabilization, which involves passing L through a local disc as indi-cated. Negative stabilization involves passing L through a similar local disc, but with the paritiesof the singularities reversed. This figure is by Etnyre and Honda.

So in terms of formulas, we have:

tb(S±(L)) = tb(L) − 1 r(S±(L)) = r(L) ± 1 (2.4)

We also note that S+(S−(L)) = S−(S+(L)). An interesting fact to note is that any twoLegendrian knots representing the same knot type become Legendrian isotopic after somenumber of positive and negative stabilizations [20].

Positive and negative destabilization of Legendrian knots is the reverse of stabilization.

2.3.5 Transversal push-offs of Legendrian knots

Let L be a Legendrian knot in a contact 3-manifold (M, ξ). Then locally, by a Darboux-typetheorem, L and ξ are contactomorphic to the x-axis and ξstd in Figure 2.3. In that figure,if we examine a thin annulus containing the x-axis but tilted slightly off of the xy-plane,that annulus has a characteristic foliation as in Figure 2.9. It is then evident that if wetake a push-off of L in one direction, we will obtain a transverse knot that is positivelyoriented with respect to the contact structure; we call this the positive transverse push-off


of L, and denote it T+(L). Similarly, a push-off in the opposite direction yields the negativetransverse push-off of L, denoted T−(L).

Figure 2.9: Shown is a local picture of an annulus containing a Legendrian knot L, and theassociated transverse push-offs T±(L). This figure is by Etnyre.

The self-linking numbers of T±(L) are given in terms of the tb and r values for L by thefollowing formula:

sl(T±(L)) = tb(L) ∓ r(L) (2.5)

Any transversal knot representative T of a knot type K can be obtained as a transversalpush-off of some Legendrian representative L.

2.3.6 Legendrian simplicity implies transversal simplicity

Motivated by the negative transverse push-off of a Legendrian knot, if L is a Legendrianknot then we define the stable Bennequin invariant s(L) to be

s(L) = tb(L) + r(L) (2.6)

So s(L) = sl(T−(L)), and s is an invariant for L up to Legendrian isotopy and positivestabilization. For a given knot type, two Legendrian knots L and L′ with the same valuesof s are said to be stably isotopic if there exists n and n′ such that Sn

+(L) = Sn′

+ (L′), wherehere Sn

+ represents n consecutive positive stabilizations. If all such Legendrian knots withthe same values of s are stably isotopic, then we say that the knot type K is stably simple.The following is then a theorem of Etnyre and Honda [18]:

Proposition 2.3.1 (Etnyre-Honda) A knot type is stably simple if and only if it istransversally simple.

Corollary 2.3.2 If a knot type is Legendrian simple, then it is transversally simple.

The converse is not true [12].

2.3.7 Legendrian mountain ranges

We conclude this section with a useful visual way of understanding the classification ofLegendrian isotopy classes for a fixed knot type. Specifically, if we fix a knot type K


in a tight contact 3-manifold (M, ξ), then we can represent Legendrian isotopy classes aspoints in the (r, tb)-plane, where r values are plotted on the horizontal axis, and tb values areplotted on the vertical axis. Since tb values are bounded above by tb(K), this plot of isotopyclasses will take the visual form of a mountain range, and in fact is called the Legendrianmountain range for K. Figure 2.10 shows the Legendrian mountain range for the (−7, 3)torus knot in (S3, ξstd); it is a Legendrian simple knot type, since at each (r, tb) value thereis a single dot, representing a single isotopy class. The mountain range is unbounded frombelow.

Figure 2.10: Shown is the Legendrian mountain range for the (−7, 3) torus knot; this is a Legendriansimple knot type. The mountain range continues to arbitrarily negative tb values. This figure is byEtnyre and Honda.

We make a few comments about Legendrian mountain ranges in general. First, arrowsdown and to the right represent positive stabilization; arrows down and to the left representnegative stabilization. Second, for Legendrian mountain ranges in (S3, ξstd) there is a well-defined map from isotopy classes at (tb, r) to isotopy classes at (tb,−r); thus Legendrianmountain ranges are symmetric about the line r = 0 (The map is 180◦ rotation about the x-axis in (R3, ker(dz−ydx))). Third, note that negative transverse push-offs of the Legendrianisotopy classes on the right-edge of the mountain range will be at sl; similarly, positivetransverse push-offs of the Legendrian isotopy classes on the left-edge of the mountain rangewill also be at sl. Another way of thinking about this is that the Legendrian mountain rangefor a knot type K will be a (possibly proper) subset of the mountain range with a singlepeak at tb = sl(K) and r = 0.

Figure 2.11 shows the Legendrian mountain range for the ((2, 3), (2, 3)) iterated torusknot in (S3, ξstd). It is a Legendrian non-simple knot type; the single dot with concentriccircles at the same value of (r, tb) represents multiple Legendrian isotopy classes.

Note that this knot type is also transversally non-simple, as the two Legendrian classesat tb = 5 and r = −2 fail to be stably isotopic.


Figure 2.11: Shown is the Legendrian mountain range for the (2, 3)-cabling of the (2, 3) torus knot;it is a Legendrian non-simple knot type, as indicated by the multiple circles at the same values of(r, tb). This figure is by Etnyre and Honda.

2.4 Cablings and iterated torus knots

2.4.1 Definition of a cabled knot type

Let K be a representative of a topological knot type, and N(K) a regular neighborhoodof K. Simple closed curves on ∂N(K) can be described as ordered pairs (P, q), where Pand q are co-prime integers, q > 0 and represents (efficient) geometric intersection with theboundary of a meridian disc for N(K), and P is the algebraic intersection of the simpleclosed curve with a longitude in some framing. If we require q > 1, then the (P, q) curveis a new knot type called the (P, q)-cabling of K, and denoted K(P,q). We will call P/q thecabling fraction, while its reciprocal q/P will be the cabling slope.

2.4.2 Two framings on ∂N(K(P,q))

Consider K(P,q) embedded on ∂N(K); if we take a small regular neighborhood N(K(P,q)),then ∂N(K(P,q)) will intersect ∂N(K) in two parallel curves, each of which intersects theboundary of a meridian disc for N(K(P,q)) once. One copy of these two parallel curves thusis a longitude for a framing on ∂N(K(P,q)); we call this framing C′. This framing is a non-standard framing; the standard framing has a longitude coming from the Seifert surface forK(P,q), and will be denoted by C. We will sometimes call C the preferred framing.

Since slopes are measured as fractions q/P , the longitude for the C framing will haveslope ∞; we will say that the longitude for the C′ framing has slope ∞′.

A convention we will be using is that meridians in the standard C framing, that is, alge-braic intersection with ∞, will be denoted by upper-case P . On the other hand, meridiansin the non-standard C′ framing, that is, algebraic intersection with ∞′, will be denoted bylower-case p.

Given a curve (P, q) on a torus ∂N(K), then there is an easy relationship between theframings C′ and C on ∂N(K(P,q)). In terms of a change of basis, we can represent slopesλ/µ as column vectors and then get from a slope λ/µ′, measured in C′ on ∂N(K(P,q)), to aslope λ/µ, measured in C, by:

2.4. CABLINGS AND ITERATED TORUS KNOTS 23

(1 Pq0 1

) (µ′

λ

)=

(µλ

)(2.7)

In other words, µ = µ′ + Pqλ; this change of basis will be justified in Section 2.4.6.

2.4.3 Definition of iterated torus knots

Iterated torus knots, as topological knot types, can be defined recursively. Let 1-iteratedtorus knots be simply torus knots (P1, q1) with P1 and q1 co-prime nonzero integers, and|P1|, q1 > 1. Here P1 is the algebraic intersection with a longitude, and q1 is the (efficient)geometric intersection with a meridian in the preferred framing for a torus representingthe unknot. Then for each (P1, q1) torus knot, take a solid torus regular neighborhoodN((P1, q1)); the boundary of this is a torus, and given a framing we can describe simpleclosed curves on that torus as co-prime pairs (P2, q2), with q2 > 1. In this way we obtain all2-iterated torus knots, which we represent as ordered pairs, ((P1, q1), (P2, q2)). Recursively,suppose the (r−1)-iterated torus knots are defined; we can then take regular neighborhoodsof all of these, choose a framing, and form the r-iterated torus knots as ordered r-tuples((P1, q1), ..., (Pr−1, qr−1), (Pr, qr)), again with Pr and qr co-prime, and qr > 1.

For ease of notation, if we are looking at a general r-iterated torus knot type, we willrefer to it as Kr; a Legendrian representative will usually be written as Lr. Note that wewill use the letter r both for the rotation number and as an index for our iterated torusknots; context will distinguish between the two uses.

2.4.4 Iterated torus knots that support ξstd

Iterated torus knots Kr are fibered knots, and thus support a contact structure associatedwith their respective open book decompositions; the contact structure associated to aniterated torus knot Kr is denoted ξKr . Hedden has shown that ξKr is isotopic to ξstd if andonly if Kr is an iterated torus knot obtained from cabling positively at each iteration, i.e.,Pi > 0 for all 1 ≤ i ≤ r [26]. Moreover, he has shown that the Bennequin inequality issharp for an iterated torus knot if and only if Pi > 0 for all 1 ≤ i ≤ r [27].

2.4.5 The quantities Ar and Br for an iterated torus knot Kr

Given an iterated torus knot type Kr = ((p1, q1), ..., (pr, qr)) where the pi’s are measured inthe C′ framing, we define two quantities which will be helpful for calculations in Chapters4 and 5. The two quantities are:

Ar :=r∑

α=1

pα

r∏

β=α+1

qβ

r∏

β=α

qβ Br :=r∑

α=1

pα

r∏

β=α+1

qβ

+

r∏

α=1

qα (2.8)


Note here we use a convention thatr∏

β=r+1

qβ := 1. Also, if we restrict to the first i

iterations, that is, to Ki = ((p1, q1), ..., (pi, qi)), we have an associated Ai and Bi. For

example, Ai :=i∑

α=1

pα

i∏

β=α+1

qβ

i∏

β=α

qβ.

Now if Kr = ((p1, q1), ..., (pr, qr)) is a general r-iterated torus knot type, with pi’smeasured in the C′ framing, we can obtain a formula for the Pi’s as measured in the standardC framing. To this end, from equation 2.8 we obtain two useful identities:

Ar = q2rAr−1 + prqr Br = qrBr−1 + pr (2.9)

Now suppose we have a ((p1, q1), ..., (pr, qr)) iterated torus knot as described above, andlet Pi be the meridians for the i-th iteration, but as measured in the standard C framing. Todetermine Pi+1, the algebraic intersection with the preferred longitude, we use the changeof basis mentioned above to obtain Pi+1 = qi+1Piqi +pi+1. We then can prove the followinglemma:

Lemma 2.4.1 Pr = qrAr−1 + pr for r ≥ 2 and Ar = Prqr for r ≥ 1.

Proof. First observe that P1 = p1 and so equation 2.8 immediately gives us A1 = P1q1.We then use induction, beginning with a base case of r = 2. From the comments abovewe have P2 = q2A1 + p2, and thus A2 = P2q2. But then inductively we can assume thatAr−1 = Pr−1qr−1, and so again by the above comments Pr = qrAr−1 + pr, and henceAr = Prqr. �

Note that as a consequence of this lemma, the change of coordinates from the C′ framing

to the C framing on ∂N(Kr) becomes left multiplication by

(1 Ar

0 1

).

2.4.6 χ(K(P,q)) in terms of χ(K), when P > 0

Let K be a topological knot type, and Σ a minimal genus Seifert surface for K. AssumeP > 0. We can then form a Seifert surface for K(P,q) by taking q copies of Σ with boundaryon ∂N(K), along with P meridian discs in N(K), and banding these together with Pqbands. The change of basis in equation 2.7 then follows. Moreover, one can see thatχ(K(P,q)) ≥ qχ(K) + P − Pq. This is in fact an equality, due to the work of Shibuya andothers [42]:

χ(K(P,q)) = qχ(K) + P − Pq (2.10)

2.4.7 χ(Kr) when Pi > 0 for all i

Lemma 2.4.2 Suppose Kr = ((P1, q1), ..., (Pr, qr)) is an iterated torus knot where Pi > 0for all i. Then −χ(Kr) = Ar − Br.

2.4. CABLINGS AND ITERATED TORUS KNOTS 25

Proof. We know that:

χ(Kr) = qrχ(Kr−1) − Prqr + Pr (2.11)

Now for a positive torus knot (P1, q1), we have χ = −A1 + B1, so we can inductivelyassume the lemma holds for Kr−1. Thus using the recursive expression we have

χ(Kr) = qrχ(Kr−1) − Prqr + Pr

= qr(−Ar−1 + Br−1) − Ar + qrAr−1 + pr−1

= −Ar + Br

�


Chapter 3

Convex surface theory and theUTP

In this chapter we review key aspects of convex surface theory, as well as background andresults concerning the uniform thickness property. For convex surface theory, our mainreferences are [22, 28, 24]; for the uniform thickness property, our references will be [17, 44].At certain points we will sketch proofs of results so as to give the reader an idea of argumentsand connections within convex surface theory.

3.1 Convex surfaces

3.1.1 Definition of a convex surface

Let S be an embedded surface, possibly with boundary, in a contact 3-manifold (M, ξ). Sis said to be convex if in a neighborhood of S there exists a vector field v such that v istransverse to S, and the flow of v preserves ξ. In other words, if φt is the one-parameterfamily of diffeomorphisms associated with v, then if x is a point in a neighborhood of S,we have (φt)∗(ξx) = ξφt(x). This neighborhood of S is often referred to as an I-invariantneighborhood, where t ∈ I = [0, 1]. The vector field v is called a contact vector field.

We will soon see that in some sense a convex surface is the generic surface in a contact3-manifold.

3.1.2 The dividing set for a convex surface

Let S be a convex surface and v its associated contact vector field. Let Γ be the set of allpoints x in S such that vx ∈ ξx. Then Γ will be an embedded multicurve in S, and is calledthe dividing set for S. The term “dividing set” is used since Γ divides S into positive andnegative regions S+ and S−, where any singularities for the characteristic foliation Sξ in S+

are positive singularities, while singularities in S− are negative.

27

28 CHAPTER 3. CONVEX SURFACE THEORY AND THE UTP

We will see in a later subsection that the dividing set Γ for a convex surface S contains allthe information needed to understand ξ in a neighborhood of S. Specifically, the followingheuristic principle will apply:

Key Principle: It is the dividing set that determines the contact structure within aneighborhhood of a convex surface.

In general, if F is any singular foliation (not necessarily the characteristic foliation) onan orientable surface S, and Γ is a disjoint union of curves on S, then we say that Γ dividesF if the following hold:

• Γ is transverse to F .

• S\Γ is the disjoint union of two surfaces S+ and S− with ∂S+ = −∂S− = Γ.

• There is a vector field u and volume form ω on S so that u is tangent to F , ±Luω > 0on S± and u|Γ points out of S+.

We can now state the following proposition which shows that for closed oriented surfacesS having characteristic foliation Sξ, the existence of a dividing set is equivalent to beingconvex:

Proposition 3.1.1 (Giroux) A closed oriented surface S in a contact 3-manifold (M, ξ)is convex if and only if the characteristic foliation Sξ is divided by a collection Γ of embed-ded simple closed curves on S. Furthermore, the dividing set of such a convex surface isdetermined by the characteristic foliation Sξ, up to an isotopy via curves transverse to Sξ.

3.1.3 Existence of closed oriented convex surfaces

Not every embedded closed oriented surface S in a contact 3-manifold (M, ξ) is convex.However, in this section we state propositions which show that convex surfaces are easyto find and in some sense can be assumed to be the generic type of surface in contact3-manifolds.

Let S be a closed oriented surface, along with a singular foliation F generated by avector field u. Then recall that u is said to be of Morse-Smale type if the following hold:

• There are finitely many singularities and closed orbits, all of which are non-degenerate.

• The α− and ω−limit set of each flow line is either a singular point or a closed orbit.

• There are no trajectories connecting hyperbolic (saddle) points.

There are then the following three results that exhibit the generic quality of convexsurfaces.

3.1. CONVEX SURFACES 29

Proposition 3.1.2 Let S be a closed oriented surface in (M, ξ). Then there is a surface S′

isotopic to, and C∞−close to, S such that the characteristic foliation S′ξ is of Morse-Smale

type.

Proposition 3.1.3 If Sξ is of Morse-Smale type, then S is convex.

Corollary 3.1.4 (Giroux) Let S be a closed oriented surface in (M, ξ). Then there is asurface S′ isotopic to, and C∞−close to, S such that S′ is convex.

3.1.4 Existence of convex surfaces with Legendrian boundary

Suppose L is a component of the boundary of a surface S, all of whose boundary componentsare Legendrian. We define the twisting number t(L, S) to be the number of counterclockwise2π twists of ξ along L, relative to the framing induced by S. In particular, if L is theboundary of a Seifert surface, then t(L, S) = tb(L).

Given this definition we have the following existence theorem for convex surfaces withLegendrian boundary.

Proposition 3.1.5 (Honda) Let S be a compact, oriented, properly embedded surface withLegendrian boundary, and assume t(L, S) ≤ 0 for all components L of ∂S. Then there existsa C∞-small perturbation (fixing ∂S) which makes S convex.

By Legendrian stabilization one can decrease the twisting number of a Legendrianboundary component; thus the above proposition guarantees the existence of many con-vex surfaces with Legendrian boundary.

In fact it is the non-positive twisting of Legendrian boundary components that charac-terizes whether S can be isotoped to be convex:

Proposition 3.1.6 (Honda) Let S be a compact, oriented, properly embedded surface withLegendrian boundary; then S can be made convex if and only if the twisting of ξ about eachboundary component is less than or equal to zero.

A similar proposition due to Kanda gives a characterization of when a closed surface Scan be made convex relative to a Legendrian curve L:

Proposition 3.1.7 (Kanda) If L is a Legendrian curve in a closed surface S, then S maybe isotoped relative to L so that it is convex if and only if t(L, S) ≤ 0. Moreover, if S isconvex, then

t(L, S) = −1

2#(L ∩ Γ) (3.1)

where #(L ∩ Γ) is the cardinality of L ∩ Γ.

If L is the Legendrian boundary of a Seifert surface Σ, the dividing set for Σ determinesthe Thurston-Bennequin number and rotation number for L:


Proposition 3.1.8 (Kanda) Suppose Σ has a single boundary component L which is Leg-endrian. Then Σ may be made convex if and only if tb(L) ≤ 0. Moreover, if Σ is convexwith dividing curves Γ, then

tb(L) = −1

2#(L ∩ Γ) (3.2)

and

r(L) = χ(Σ+) − χ(Σ−) (3.3)

where Σ± are as in the definition of convexity.

3.1.5 Giroux Flexibility

We mentioned above that the key idea for dividing curves is that it is the dividing set (not theexact characteristic foliation) which encodes the essential contact topological informationin a neighborhood of a convex surface S. The following fundamental theorem of Giroux,called the Giroux Flexibility Theorem, makes this precise:

Theorem 3.1.9 (Giroux) Let S be a closed convex surface or a compact convex surfacewith Legendrian boundary, and with characteristic foliation Sξ, contact vector field v, anddividing set Γ. If F is another singular foliation on S divided by Γ, then there is an isotopyφt, t ∈ [0, 1], of S such that φ0(S) = S, φ1(S)ξ = F , the isotopy is fixed on Γ, and φt(S) istransverse to v for all t.

In short, any foliation divided by Γ can be realized as a characteristic foliation for Safter a small isotopy.

A consequence of Giroux Flexibility is the following proposition, which essentially saysthat homotopically trivial dividing curves yield overtwisted discs:

Proposition 3.1.10 (Giroux) If S 6= S2 is a convex surface in a contact 3-manifold(M, ξ), then S has a tight neighborhood if and only if Γ has no homotopically trivial curves.If S = S2, then S has a tight neighborhood if and only if #Γ = 1.

Here, #Γ is the number of components in the dividing set.

Giroux elimination

There is another lemma that is related to Giroux Flexibility but which is more specific; itsays that adjacent elliptic and hyperbolic singularities of the same sign in a characteristicfoliation can be eliminated; this is called Giroux elimination. Here, adjacent means there isa non-singular arc joining the two singularities.

Lemma 3.1.11 (Giroux) Any two adjacent elliptic and hyperbolic singularities of thesame sign in Sξ may be removed by a local C0-isotopy of S.


Legendrian realization

The following lemma shows that most closed curves on a surface can be made to be Legen-drian after an isotopy of the surface:

Lemma 3.1.12 (Honda, Kanda) If C is a closed curve on a convex surface S with Leg-endrian boundary, and such that C is transverse to Γ with C ∩ΓS 6= ∅, then there exists anisotopy φs, s ∈ [0, 1] so that

1. φ0 = id,

2. φs(S) are all transverse to the contact vector field,

3. φs(S) are all convex,

4. φ1(ΓS) = Γφ1(S),

5. φ1(C) is Legendrian.

3.1.6 Standard form for convex tori

We now begin to focus primarily on convex tori and cabled knot types.

Legendrian divides and rulings

By Proposition 3.1.10, any convex torus T 2 in a tight contact 3-manifold will have 2nparallel dividing curves with some rational slope s. By Giroux Flexibility, the characteristicfoliation T 2

ξ can be assumed to have the following standard form:

• 2n simple closed curves of singularities, thought of as parallel push-offs of the compo-nents of Γ. These curves of singularities will alternate between positive and negativecurves of singularities as one moves between components of T 2

+ to T 2−. These curves

of singularities are Legendrian curves called Legendrian divides. For a given convextorus, the slope of the Legendrian divides, as well as the number of divides, is fixed.

• A one-parameter family of Legendrian curves all of the same rational slope r, andcalled Legendrian rulings. By Giroux Flexibility, r may be any rational slope otherthan s (the slope of the divides).

For our purposes, what is extremely important is that by using Legendrian divides andrulings, we can obtain all cabling knot types as Legendrian knots on any convex torus.


The twisting t, and tb and r, for Legendrian rulings and divides

Suppose N is a solid torus representing some knot type K, where ∂N is convex. If L is aLegendrian (P, q)-cabling on ∂N , then if we define t to be the twisting of the contact planesalong L with respect to the C′ framing on ∂N(L), we have the following equation [17]:

tb(L) = Pq + t(L) (3.4)

Observe that t(L) is also the twisting of the contact planes with respect to the framinggiven by ∂N , and so is equal to −(1/2)#(L ∩ Γ∂N ). The maximal twisting number withrespect to this framing will be denoted by t. Note that if L is a Legendrian divide, we havet(L) = 0.

We then have the following lemma that shows how to compute tb for Legendrian dividesand rulings on a convex torus ∂N(K) with two dividing curves.

Lemma 3.1.13 (Etnyre-Honda)

1. Suppose L(P,q) is a Legendrian divide and slope(Γ∂N(K)) = q/P . Then tb(L(P,q)) =Pq.

2. Suppose L(P,q) is a Legendrian ruling and slope(Γ∂N(K)) = q′/P ′. Then tb(L(P,q)) =Pq − |Pq′ − qP ′|.

Next we explain how to compute the rotation number r(L(P,q)).

Lemma 3.1.14 (Etnyre-Honda) Let D be a convex meridional disc of N(K) with Leg-endrian boundary on a contact-isotopic copy of the convex surface ∂N(K), and let Σ be aconvex Seifert surface with Legendrian preferred longitude boundary on a contact-isotopiccopy of ∂N(K). (Here, the isotopic copies of ∂N(K) are copies inside an I-invariantneighborhood of ∂N(K), obtained by applying Giroux Flexibility.) Then

r(L(P,q)) = P · r(∂D) + q · r(∂Σ) (3.5)

Definition of boundary slope and intersection boundary slope

If N is a solid torus embedded in a tight contact 3-manifold (M, ξ) such that ∂N is convex,then we call the slope of the dividing curves on ∂N the boundary slope, which we will usuallythink of as a fraction in lowest terms.

Let N be a solid torus with convex boundary in standard form, and with slope(Γ∂N ) =a/b in some framing. If |2b| is the geometric intersection of the dividing set Γ with alongitude ruling in that framing, then we will call a/b the intersection boundary slope. Notethat when we have an intersection boundary slope a/b, then 2gcd(a, |b|) is the number ofdividing curves.


3.1.7 Edge-rounding

Suppose we have two convex surfaces S1 and S2 which intersect transversely along a commonLegendrian boundary component. Then their dividing curves will interleave as in part (a)of Figure 3.1; after rounding the edge, the dividing curves on the resulting surface will beas in part (b). In short, the edge-rounding satisfies a left-hand rule; if the fingers of one’sleft hand curl along the two intersecting surfaces, the left thumb points in the direction inwhich dividing curves join up.

Figure 3.1: Part (a) shows two convex surfaces intersecting transversely along a Legendrian bound-ary component; the heavier black line segments are dividing curves. Part (b) shows the dividingcurves on the resulting surface after edge-rounding.

3.1.8 Bypasses

Let S be a convex surface; a bypass for S is an oriented embedded half-disc D with Legen-drian boundary, satisfying the following conditions (refer to Figure 3.2):

• ∂D is the union of two arcs γ1, γ2 which intersect at their endpoints.

• D intersects S transversely along γ1.

• D (or D with opposite orientation) has the following singularities in its characteristicfoliation:

– positive elliptic singularities at the endpoints of γ1.

– one negative elliptic singularity on the interior of γ1.

– only positive singularities along γ2, alternating between elliptics and hyperbolics.

• γ1 intersects ΓS exactly at three points, and these three points are the elliptic pointsof γ1.


Figure 3.2: Shown is a bypass attached along a Legendrian ruling curve; it represents a destabi-lization of that Legendrian ruling. This figure is by Honda.

We will often call the arc γ2 a bypass for S or a bypass for γ1. We define the sign ofa bypass to be the sign of the half-elliptic point at the center of the half-disc, where theorientation of the half-disc agrees with the orientation of its Legendrian boundary arc onS.

Notice that if γ1 is a segment of a Legendrian knot L, then a bypass represents a desta-bilizing disc for L, either positive or negative depending on the sign of the bypass. Noticealso that the half-disc associated with a bypass has a single boundary-parallel dividing curvewith endpoints on γ1; we shall see that in fact such a boundary-parallel dividing curve willbe our usual way of finding bypasses.

Finally, note that a bypass half-disc can be thought of as half of an overtwisted disc(though in a tight contact 3-manifold, the other half of such an overtwisted disc can neverexist).

3.1.9 How bypasses can change dividing sets

A bypass off of a general convex surface

The following lemma gives a local picture of how passing through a bypass changes thedividing set of a convex surface.

Lemma 3.1.15 (Honda) Assume D is a bypass for a convex S. Then there exists aneighborhood of S ∪ D diffeomorphic to S × [0, 1], such that Si = S × {i}, i = 0, 1, areconvex, S × [0, ǫ] is I-invariant, and ΓS1 is obtained from ΓS0 by performing the bypassattachment operation depicted in Figure 3.3 in a neighborhood of the attaching Legendrianarc γ1.

Proof. Figure 3.4 shows the dividing set on S1; edge-rounding then establishes the lemma.�


Figure 3.3: Part (a) shows γ1 intersecting ΓS0three times; part (b) shows the resulting ΓS1

afterpassing through the bypass. This figure is by Honda.

Figure 3.4: Shown is S1 with dividing curves. This figure is by Honda.


A bypass off of a standard form convex torus

Let T 2 be a convex torus in standard form; the slope of the Legendrian divides will bes, and we assume that there is a bypass attachment with boundary along a Legendrianruling of slope r. After changing framings via SL(2, Z) we may assume s = 0 and thatr 6= 0 is rational. Furthermore, we can normalize the Legendrian rulings via an element(

1 m0 1

)∈ SL(2, Z) so that −∞ ≤ r ≤ −1. We then have the following lemma which

establishes how bypasses change the dividing sets of standard form convex tori.

Lemma 3.1.16 (Honda) Assume a bypass D is attached to T 2 with slope s = 0, along aLegendrian ruling curve of slope r with −∞ ≤ r ≤ −1. Then there exists a neighborhoodT 2 × I of T 2 ∪D with ∂(T 2 × I) = T1 − T0, such that ΓT0 = ΓT 2 and ΓT1 will be as follows,depending on whether #ΓT0 = 2 or #ΓT0 > 2:

1. If #ΓT0 > 2, then s1 = s0 = 0, but #ΓT1 = #ΓT0 − 2.

2. If #ΓT0 = 2, then s1 = −1 and #ΓT1 = 2.

Here si is the boundary slope of Ti.

Proof. This follows from the above bypass attachment lemma and Figure 3.5. Note thatin the case #Γ = 2, a bypass attachment effectively performs a positive Dehn twist.

Figure 3.5: The top row shows the effect of passing through a bypass on a convex torus with morethan 2 dividing curves; the slope stays the same, but the number of dividing curves decreases by 2.The bottom row shows that if one begins with 2 dividing curves and then passes through a bypass,the slope changes. This figure is by Honda.

�


Using the Farey tessellation to calculate slope changes

We can interpret the effect of bypasses on slopes of convex tori using the Farey tessellationof the hyperbolic disc and the action of SL(2, Z). To this end, let the hyperbolic unitdisc be H

2 ={(x, y)|x2 + y2 ≤ 1

}. We will label points on ∂H

2 as follows (refer to Figure3.6). Begin by labeling (1, 0) as 0 = 0/1, and (−1, 0) as ∞ = 1/0. We then inductivelylabel points on ∂H

2 for y > 0 in the following manner: Suppose we have already labeled∞ ≥ P/q ≥ 0 (P, q relatively prime) and ∞ ≥ P ′/q′ ≥ 0 such that (P, q), (P ′, q′) form aZ-basis for Z

2. Then, halfway between P/q and P ′/q′ along ∂H2 on the shorter arc (one for

which y > 0), we label (P +P ′)/(q + q′). We then connect two points P/q and P ′/q′ on theboundary if the corresponding shortest integral vectors form an integral basis for Z

2. SeeFigure 3.6. We then reflect across the x-axis and negate to form the tessellation for y < 0.

Figure 3.6: Shown is the Farey tessellation of the hyperbolic disc; vertices joined by arcs representa Z-basis for Z

2. This figure is by Honda.

By transforming the above lemma via SL(2, Z), we obtain the following rephrasing inmore invariant language.

Lemma 3.1.17 (Honda) Let T be a convex torus with #Γ = 2 and slope s. If a bypassis attached off of a ruling curve with slope r, then the resulting convex torus T ′ will have#Γ = 2 and slope s′ obtained as follows: Take the arc [r, s] ⊂ ∂H

2 obtained by startingfrom r and moving counterclockwise until we hit s. On this arc, let s′ be the point which isclosest to r and has an edge from s′ to s.


3.1.10 Isotopy discretization

Bypasses are the generic way to isotop one convex surface to another. In particular, wehave the following proposition:

Proposition 3.1.18 (Honda) Consider S × I, where the S × {i} = Si are convex fori = 0, 1. Then one can obtain S1 from S0 through a finite sequence Sti, with t0 = 0 <t1 < · · · < tn = 1, such that each Sti is convex, and such that Sti+1 is obtained from Sti bypassing through a bypass.

3.1.11 How to find bypasses: convex discs and annuli

In this subsection we present two common ways to find bypasses off of convex tori. The ideais to find Legendrian curves on such tori that also form boundary components for eitherdiscs or annuli, and then determine that there must be boundary-parallel dividing curves,and hence bypasses.

Lemma 3.1.19 (Honda) Let D be a convex disc with Legendrian boundary inside a tightcontact 3-manifold, and with t(∂D, D) < 0. Then every component of Γ is an arc whichbegins and ends on ∂D, and if t(∂D) < −1 there exists a bypass along ∂D.

Proof. There cannot be a simple closed dividing curve, as this would result in an over-twisted disc; thus all dividing curves are arcs. A simple count of intersections of ∂D withΓ now shows that if t < −1 we must have at least two boundary-parallel dividing curves,in which case one is a bypass. �

Now if S1× [0, 1] is a convex annulus with Legendrian boundary inside of a tight contact3-manifold, then again all dividing curves must be arcs that have endpoints either on thesame boundary component, or on different boundary components. The following ImbalancePrinciple then gives a criterion by which the existence of boundary-parallel dividing curvesare assured.

Lemma 3.1.20 (Imbalance Principle, Honda) Let S1 × [0, 1] be convex with Legen-drian boundary inside a tight contact 3-manifold. If t(S1 × {0}) < t(S1 × {1}) ≤ 0, thenthere exists a bypass along S1 × {0}.

Proof. By a count of intersections of the two boundary components with Γ, it is evidentthat there must be a boundary-parallel dividing curve along S1 × {0}; an innermost oneyields a bypass. �

If the dividing set on a convex annulus consists solely of arcs with one endpoint oneach boundary component (meaning no boundary-parallel dividing curves, and hence nobypasses), such a convex annulus is said to be standard convex.


3.1.12 Relative Euler class

Let (M, ξ) be a tight contact 3-manifold with convex boundary ∂M . Assume ξ|∂M istrivializable, and choose a nowhere zero section s of ξ on ∂M . Then we may form the relativeEuler class e(ξ, s) ∈ H2(M, ∂M ; Z); specifically, consider the following exact sequence:

H1(∂M) → H2(M, ∂M) → H2(M) → H2(∂M) (3.6)

e(ξ, s) 7→ e(ξ) 7→ 0

This implies that a nonzero section s of ∂M allows for a lift of e(ξ) to e(ξ, s). Therelative Euler class can be evaluated as follows:

Proposition 3.1.21 (Honda) Let (M, ξ) be a contact manifold with convex boundary. Fixa nonzero section s of ξ|∂M .

1. If Σ ⊂ M is a closed convex surface with positive (resp. negative) region Σ+ (resp.Σ−) divided by ΓΣ, then 〈e(ξ, s), Σ〉 = χ(Σ+) − χ(Σ−).

2. If Σ ⊂ M is a compact convex surface with Legendrian boundary on ∂M and re-gions Σ+ and Σ−, and s is homotopic to s′ which coincides with γ for every orientedconnected component γ of ∂Σ, then 〈e(ξ, s), Σ〉 = χ(Σ+) − χ(Σ−).

Let T be a standard form convex torus. Then a nonzero section s of ξ|T is given by thetangent field of the rulings. We have the following lemma:

Lemma 3.1.22 (Honda) Let (M, ξ) be a tight contact manifold with convex boundaryconsisting of tori. Then the relative Euler class 〈e(ξ, s), Σ〉 is independent of the slope ofthe Legendrian rulings, if s is given by the tangent field of the rulings.

We can now explain how to compute relative Euler classes for two spaces of inter-est; assume ξ is tight. For S1 × D2 with convex boundary, we use Giroux flexibilityto make the Legendrian rulings meridian curves, and compute the relative Euler classof a meridional convex disc Σ by taking 〈e(ξ, s), Σ〉 = χ(Σ+) − χ(Σ−). We have thate(ξ, s) ∈ H2(M, ∂M ; Z) = H1(M ; Z) = Z, so evaluation on a single meridional disc com-pletely determines the relative Euler class.

Similarly, for T 2 × I, we modify the boundary so the Legendrian rulings have the sameslope r for both T 2 ×{0} and T 2 ×{1}. Take a convex annulus A = γ × I with Legendrianboundary, where γ is a closed curve with slope r. If we compute 〈e(ξ, s), A〉 = χ(A+)−χ(A−)for two annuli of two different slopes, then this will completely determine the elemente(ξ, s) ∈ H1(T

2 × I; Z) = H1(T2; Z) = Z

2.


3.1.13 Classification of tight contact structures for B3, S1×D2, and T 2×I

In this subsection we will present classification results for tight contact structures on 3-manifolds with convex boundary, in particular for B3, S1 × D2, and T 2 × I. For S1 × D2

and T 2 × I we will not present complete classification results, but rather focus on what wewill need for the rest of this work.

3-ball classification

The following is a fundamental theorem [14]:

Theorem 3.1.23 (Eliashberg) Assume there exists a contact structure ξ on a neighbor-hood of ∂B3 which makes ∂B3 convex with #Γ = 1. Then there exists a unique extensionof ξ to a tight contact structure on B3, up to isotopy which fixes the boundary.

Solid tori classification

Let L be a Legendrian knot with twisting n ∈ Z in some framing. Then a standard neigh-borhood of L is a neighborhood N(L) so that ∂N(L) is convex with slope(Γ) = 1/n and#Γ = 2. This can be seen by simply noticing that as the contact planes twist along L, theytrace out two dividing curves on ∂N(L). The contact structure on a standard neighborhoodof a Legendrian knot is universally tight.

The following proposition shows that any tight solid torus having convex boundary withslope(Γ) = 1/n and #Γ = 2 is contactomorphic to a standard neighborhood of a Legendrianknot with twisting n. (Recall that for a solid torus with convex boundary, we call the slopeof the dividing curves on the boundary the boundary slope.)

Proposition 3.1.24 (Honda) There exists a unique tight contact structure on S1 × D2

with a fixed convex boundary with #Γ∂(S1×D2) = 2 and boundary slope(Γ∂(S1×D2)) = 1/n;this tight contact structure is isotopic to the standard neighborhood of a Legendrian curvewith twisting n.

Proof. The idea is straightforward. First, by Eliashberg’s classification theorem for tightcontact structures on the 3-ball, the tight contact structure inside S1 × D2 is determinedby the characteristic foliation on the convex torus boundary and a convex meridian disc.For the boundary conditions in this proposition, the convex meridian disc is forced to havea single dividing curve dividing it into two half discs, D+ and D−. Thus there is only onetight contact structure possible. �

This proof illustrates a key principle which is worth stating:

Key Principle: For a tight S1 × D2 with a fixed convex boundary, it is the dividing setof a convex meridian disc which determines the contact structure.


If D is a convex meridional disc for a tight S1×D2 with convex boundary, depending onthe number of intersections of Γ∂(S1×D2) with ∂D, there will be many possible configurationsfor dividing curves on D, and hence many possible tight contact structures on S1×D2 withthe given boundary conditions. However, the next proposition shows that, given the rightconditions, there are only two universally tight contact structures.

Proposition 3.1.25 (Honda) There are exactly two tight contact structures on N =S1 × D2 with #Γ∂N = 2 and boundary slope s = −p/q < −1 which are universally tight.Specifically, these two universally tight contact structures have convex meridian discs whosedividing sets consist solely of boundary-parallel bypasses all of the same sign, and are thusdiffeomorphic via −id.

Proof. The following is a brief sketch of the proof. These two contact structures areuniversally tight since they can be embedded into (T 3, ξ1), which is a tight contact 3-torusthat is also universally tight (see [28]). If a convex meridian disc has half-discs of differentsigns, then one shows that in a finite cover one can find an overtwisted disc. �

Notice that a solid torus with convex boundary in a tight contact 3-manifold cannothave a dividing curve of slope 0 (i.e., a meridian dividing curve), since this would yield anovertwisted disc. Thus if we refer back to Figure 3.6, then for boundary slopes for solid toriin tight contact 3-manifolds, there is essentially a singularity at (1, 0) = 0/1. However, thefollowing proposition shows that other boundary slopes are prevalent in a systematic way.

Proposition 3.1.26 (Honda) Let N be a solid torus with convex boundary in a tight 3-manifold (M, ξ). Suppose s is the boundary slope for N , and s′ is a slope between s and 0,on the arc obtained by moving counterclockwise from s to 0 on the boundary of the Fareytessellation. Then there exists N ′ ⊂ N where the boundary slope of N ′ is s′.

Thickened tori classification

Consider (T 2 × I, ξ), where Ti = T 2 × {i}, i = 0, 1, are convex tori. We first introducesome terminology. If we fix a framing for the Ti, then an annulus in T 2× I joining meridiancurves on the Ti will be called a horizontal annulus, whereas an annulus joining longitudeson the Ti will be called a vertical annulus.

If we define the two boundary slopes on the two convex tori to be si for i = 0, 1, wemay assume (after possible re-labelling) that the interior of the arc on the Farey tessellationthat is obtained by moving counterclockwise from s1 to s0 does not contain 0. We then saythat ξ is minimally twisting if every slope st for a T 2 ×{t}, t ∈ (0, 1), is such that st lies onthe above arc in the Farey tessellation from s1 to s0. The contact structure ξ is said to benonrotative if it is minimally twisting and s0 = s1.

We then call (T 2 × I, ξ) a basic slice if

• ξ is tight.


• Ti are convex and #ΓTi= 2 for i = 0, 1.

• The minimal integral representatives of Z2 corresponding to si form a Z-basis of Z

2

(i.e., the si are joined by an arc in the Farey tessellation).

• ξ is minimally twisting.

After a diffeomorphism of T 2, we may assume that a basic slice has boundary slopess1 = −1 and s0 = 0 (recall that our manifold is T 2×I and thus a slope 0 dividing curve doesnot bound a disc, so the last statement does not imply overtwisting). Then the followingclassifies tight contact structures on basic slices:

Proposition 3.1.27 (Honda) Let ΓTi, i = 0, 1, satisfy #ΓTi

= 2 and s1 = −1, s0 = 0.Then there exist two tight contact structures on T 2 × I, and both are universally tight; theyare diffeomorphic via −id and the Poincare duals to the relative Euler classes are given by±(0, 1) ∈ H1(T

2; Z).

For more general thickened tori, we have the following classification of universally tightcontact structures:

Proposition 3.1.28 (Honda) There are exactly two universally tight contact structureson T 2 × I with minimal twisting, #Γi = 2, and boundary slopes s1 = −p/q, s0 = 0, wherep > q > 0 are positive integers. Specifically, they have horizontal annuli with bypasses allof the same sign, and thus are diffeomorphic via −id.

Note that the last statement in this proposition implies the following: If we decomposeT 2 × I as a layering of parallel basic slices, then in order for the T 2 × I to be universallytight, there must be no mixing of sign at the level of the Poincare duals to the relativeEuler classes associated with those basic slices.

We also have the following two propositions:

Proposition 3.1.29 (Honda) Let T 2 × I be embedded in a tight contact 3-manifold, withboundary slopes si, i = 0, 1. Then we can find convex tori parallel to T 2 × {i} with anyboundary slope s in [s1, s0].

Proposition 3.1.30 (Honda) If T 2 × I is a tight basic slice, with s ∈ (s1, s0), then thereexists a linearly foliated torus T parallel to T 2 × {i}, and such that every convex surfaceT ′ in standard form with slope s, after contact isotopy, is transverse to T , and T ∩ T ′ isexactly the union of the Legendrian divides of T ′.

3.1.14 Twist number lemma

We use the following lemma extensively, and thus it merits its own subsection:

3.2. THE UNIFORM THICKNESS PROPERTY FOR KNOTS 43

Lemma 3.1.31 (Honda) Let (M, ξ) be a tight contact 3-manifold with a fixed framing,and L a Legendrian knot with twisting n. Let r be the slope of a Legendrian ruling curveon ∂N(L). If there exists a bypass attached along this ruling curve, and 1/r ≥ (n+1), thenthere exists a Legendrian curve with larger twisting number isotopic (but not Legendrianisotopic) to L.

Proof. The proof is simply an application of Lemma 3.1.17. �

3.2 The uniform thickness property for knots

The two main references for this section are [17, 44].

3.2.1 Definition of the contact width of a knot

Let (M, ξ) be a tight contact 3-manifold. For a fixed (null-homologous) topological knottype K, define the contact width of K to be

w(K) = sup1

slope(Γ∂N )(3.7)

In this equation the N are solid tori having representatives of K as their cores. Slopesare measured using the preferred framing coming from a Seifert surface for K. Any knottype K satisfies the inequality tb(K) ≤ w(K) ≤ tb(K)+1, which is essentially an applicationof the twist number lemma above.

3.2.2 Definition of the uniform thickness property (UTP)

A knot type K satisfies the uniform thickness property (UTP) if the following hold:

1. tb(K) = w(K).

2. Every solid torus N representing K is contained in (or can be thickened to) a standardneighborhood of a tb Legendrian knot.

We note that condition 2 actually implies condition 1; however, to prove that knotssatisfy the UTP, it is often useful to first prove 1, and then prove 2.

At first glance, this definition seems technical and fairly restrictive. However, we willsee that in the class of iterated torus knots in S3, satisfying the UTP is in fact a fairlycommon condition; furthermore, the failure or satisfaction of the UTP exactly aligns witha fundamental contact topological property of an iterated torus knot, namely whether ornot it supports the standard contact structure.


3.2.3 Ways to fail the UTP

There are three ways to fail the UTP. Specifically, a knot K will fail the UTP if either ofthe following occur:

1. tb(K) < w(K).

2. There exists a solid torus N representing K with boundary slope 1/tb, #Γ∂N > 2,and if N ′ ⊃ N , then #Γ∂N ′ > 2.

3. There exists a solid torus N representing K with boundary slope s 6= 1/tb, and suchthat if N ′ ⊃ N , then its boundary slope is s′ = s.

A solid torus in item 3 fails to thicken, or is called a non-thickenable; we will sometimesrefer to such a solid torus as non-thickenable even if s = 1/tb, though such a torus willusually be trivially non-thickenable if we know that w(K) = tb(K).

By the classification of tight solid tori and thickened tori, thickening always changesboundary slopes in a clockwise direction around the boundary of the Farey tessellation.Also, the thickened tori that result from a thickening of a solid torus are always minimallytwisting.

The unknot in (S3, ξstd) fails the UTP by criterion 1, since if U is the unknot, w(U) = 0while tb(U) = −1. However, there are no non-thickenable solid tori represented by theunknot.

3.2.4 Definition of the lower width of a knot

Tosun has introduced the following useful definition. For a fixed topological knot type K,define the lower width of K to be

lw(K) = inf1

slope(Γ∂N )(3.8)

where here the N are non-thickenable solid tori having representatives of K as their cores(in which we include non-thickenables with boundary slope 1/tb), and slopes are measuredusing the preferred framing coming from a Seifert surface for K. It is immediate thatlw(K) ≤ w(K), and it is clear that K satisfies the UTP if and only if tb(K) = w(K) = lw(K)and no solid torus N is as in criterion 2 above.

3.2.5 Results concerning the UTP

Here we list structure results for the UTP in (S3, ξstd), due to Etnyre and Honda. The firstresult shows that knots satisfying the UTP exist:

Theorem 3.2.1 (Etnyre-Honda) Negative torus knots satisfy the UTP.

The second result shows that non-thickenable solid tori exist:

3.2. THE UNIFORM THICKNESS PROPERTY FOR KNOTS 45

Theorem 3.2.2 (Etnyre-Honda) The (2, 3) torus knot fails the UTP; in particular, thereexist non-trivial non-thickenable solid tori representing the (2, 3) torus knot.

The third result shows that the UTP is closed under connected sums:

Theorem 3.2.3 (Etnyre-Honda) If two knot types K1 and K2 satisfy the UTP, thentheir connected sum K1#K2 satisfies the UTP.

The fourth result shows that the UTP is closed under cabling, provided the cablingfraction is less than the contact width:

Theorem 3.2.4 (Etnyre-Honda) If a knot type K satisfies the UTP, then the cablingK(P,q) satisfies the UTP, provided P/q < w(K).

In Chapter 4, we will both use these theorems, and build upon them.

3.2.6 Results concerning the UTP and simplicity

In this subsection we list results that connect aspects of the UTP, contact width, and lowerwidth, to questions surrounding the simplicity or non-simplicity of cabled knot types; theambient contact manifold is (S3, ξstd). The first two results are due to Etnyre and Honda;the first shows that satisfaction of the UTP is sufficient to preserve simplicity after cabling:

Theorem 3.2.5 (Etnyre-Honda) Let K be a knot type which is Legendrian simple andsatisfies the UTP; then K(P,q) is Legendrian simple.

The second result is a partial inverse to the above result. This second result showsthat the non-thickenables representing the (2, 3) torus knot are sufficient to guarantee theexistence of transversally non-simple cablings.

Theorem 3.2.6 (Etnyre-Honda) The (2, 3)-cabling of the (2, 3) torus knot is transver-sally non-simple.

The next two theorems are due to Tosun, and show that despite the possible existenceof non-thickenables representing a knot type K, if K is Legendrian simple there can still besimple cablings:

Theorem 3.2.7 (Tosun) Let K be a knot type which is Legendrian simple with w(K) ∈ Z.Then the cabling K(P,q) is Legendrian simple, provided P/q > w(K).

Theorem 3.2.8 (Tosun) Let K be a knot type which is Legendrian simple with lw(K) ∈ Z.Then the cabling K(P,q) is Legendrian simple, provided P/q < lw(K).

We will both use and expand upon these four theorems in Chapter 5, where we establishfamilies of both simple and non-simple iterated torus knots.


Chapter 4

Results: UTP cabling andclassification theorems

In this chapter we establish theorems that describe how both satisfaction and failure ofthe UTP behave under cabling, and also provide a complete UTP classification for iteratedtorus knots; the ambient contact manifold is (S3, ξstd).

4.1 Cabling theorems for the UTP

In this section we focus on understanding how the UTP behaves under the cabling operation.

4.1.1 Cabling preserves the UTP

Theorem 4.1.1 If K satisfies the UTP, then K(P,q) satisfies the UTP.

Proof. By Theorem 3.2.4, we know that (P, q) cables satisfy the UTP, provided P/q <w(K). Thus we only need to look at the case where P/q > w(K). From the proof ofTheorem 3.2.5 in [17], we know that t(K(P,q)) < 0, and that K(P,q) achieves tb(K(P,q)) as aLegendrian ruling curve on a convex torus with boundary slope 1/w(K) and two dividingcurves. (We note that neither of these statements uses the Legendrian simplicity hypothesisin the statement of Theorem 3.2.5.)

To prove that K(P,q) satisfies the UTP, it suffices to show that any solid torus N(P,q)

representing K(P,q) thickens to a standard neighborhood of a Legendrian knot at tb(K(P,q)).So given a solid torus N(P,q), let A be a convex annulus connecting ∂N(P,q) to itself, with∂A being two ∞′ rulings so that ∂N(P,q)\∂A consists of two annuli, one of which, along

with A, bounds a solid torus N representing K with N ⊃ N(P,q). Now since K satisfies

the UTP, N can be thickened to a standard neighborhood of a Legendrian knot at tb(K),which we call N . See part (a) in Figure 4.1.

We now let L be a Legendrian core curve representing K in N\N(P,q), and let A be aconvex annulus joining ∂N to ∂N(L) inside N\N(P,q), with boundary components (P, q)

47

48 CHAPTER 4. RESULTS: UTP CABLING AND CLASSIFICATION THEOREMS

Figure 4.1: Shown is a meridional cross-section of N . The larger torus in gray is N(P,q); the smallertorus in gray is N(L).

Legendrian rulings. See part (b) in Figure 4.1. We may assume that we have topologicallyisotoped L so that the Thurston-Bennequin number is maximized over all such topologicalisotopies for the space N\N(P,q). N(L) will have dividing curves of slope 1/m in C, where

m ∈ Z. We claim that in fact m = tb(K). For if m < tb(K), then by the ImbalancePrinciple (Lemma 3.1.20), there must exist bypasses on the ∂N(L)-edge of A, since the∂N -edge of A is at maximal twisting. But such a bypass would induce a destabilization ofL, thus increasing its tb by one – this is an application of the twist number lemma (Lemma3.1.31). To satisfy the conditions of this lemma, we are using the fact that P/q > w(K).Thus m = tb(K) and A is standard convex.

Finally, note that now N(P,q) thickens to N(P,q) = N\(N(A)∪N(L)). We can calculate

the boundary slope of N(P,q). We choose (P ′, q′) to be a curve on N and N(L) such thatPq′ − P ′q = 1, and we change coordinates to a basis C′′ via the map ((P, q), (P ′, q′)) 7→((0, 1), (−1, 0)). Under this map we obtain

slope(Γ∂N ) = slope(Γ∂N(L)) =q′w(K) − P ′

qw(K) − P(4.1)

We then obtain in the C′ framing, after edge-rounding, that

slope(Γ∂N(P,q)

) =q′w(K) − P ′

qw(K) − P−

q′w(K) − P ′

qw(K) − P+

1

qw(K) − P

=1

qw(K) − P=

1

t(K(P,q))(4.2)

Hence the boundary slope of N(P,q) must be 1/tb(K(P,q)) with two dividing curves in thestandard C framing. Thus K(P,q) satisfies the UTP. �

4.1. CABLING THEOREMS FOR THE UTP 49

Figure 4.2: Shown is the region R bounded by T2 − T1.

4.1.2 Cabling preserves Legendrian simplicity and the UTP

We have the following immediate corollary, obtained by combining Theorem 4.1.1 andTheorem 3.2.5:

Corollary 4.1.2 If K is Legendrian simple and satisfies the UTP, then K(P,q) is Legen-drian simple and satisfies the UTP.

4.1.3 Cablings less than integral lower width satisfy the UTP

Theorem 4.1.3 Let K be a knot type with lw(K) ∈ Z. If P/q < lw(K), then K(P,q)

satisfies the UTP.

Proof. We first examine representatives of K(P,q) at tb. Since there exists a convextorus representing K with Legendrian divides that are (P, q) cablings (inside of the solidtorus representing K with slope(Γ) = 1/lw(K)) we know that tb(K(P,q)) ≥ Pq. To show

that tb(K(P,q)) = Pq, we show that t(K(P,q)) = 0 by showing that the contact width

w(K(P,q), C′) = 0, since this will yield tb(K(P,q)) ≤ w(K(P,q)) = Pq. So suppose, for contra-

diction, that some N(P,q) has convex boundary with slope(Γ∂N(P,q)) = s > 0, as measured

in the C′ framing, and two dividing curves. After shrinking N(P,q) if necessary, we mayassume that s is a large positive integer. Then let A be a convex annulus from ∂N(P,q) toitself having boundary curves with slope ∞′. Taking a neighborhood of N(P,q) ∪ A yieldsa thickened torus R with boundary tori T1 and T2, arranged so that T1 is inside the solidtorus N representing K bounded by T2. See Figure 4.2.

Since lw(K) ∈ Z, there exists an oriented diffeomorphism of ∂N(K) that takes (lw(K), 1)to (0, 1) and fixes (1, 0). Under this change of coordinates, (P, q) becomes a negative ca-bling. We can then assume in the argument that follows, without loss of generality, that


lw(K) = 0 and P < 0. The proof then becomes nearly identical to the proof that negativetorus knots satisfy the UTP (i.e., the proof of Theorem 3.2.1 in [17]).

Now there are no boundary parallel dividing curves on A, for otherwise, we could passthrough the bypass and increase s to ∞′, yielding excessive twisting inside N(P,q). HenceA is in standard form, and consists of two parallel nonseparating arcs. We now choosea new framing C′′ for N where (P, q) 7→ (0, 1); then choose (P ′′, q′′) 7→ (1, 0) so thatP ′′q − q′′P = 1 and such that slope(ΓT1) = −s and slope(ΓT2) = 1. As mentioned in thethird paragraph of the proof of Theorem 1.2 in [17], this is possible since ΓT1 is obtainedfrom ΓT2 by (s + 1) right-handed Dehn twists. Then note that in the C′ framing, we havethat q/P > slope(ΓT2) = (q′′ + q)/(P ′′ + P ) > q′′/P ′′, and q/P and q′′/P ′′ are connectedby an arc in the Farey tessellation of the hyperbolic disc. Thus, since −∞ is connected byan arc to 0/1 in the Farey tessellation, we must have that (q′′ + q)/(P ′′ + p) < 0 = lw(K).Thus we can thicken N to a standard neighorhood with slope(Γ) = −∞. Then, just as inClaim 4.2 in [17], we have the following:

(i) inside R there exists a convex torus parallel to Ti with slope q/P ;

(ii) R can thus be decomposed into two layered basic slices;

(iii) the tight contact structure on R must have mixing of sign in the Poincare duals ofthe relative half-Euler classes for the layered basic slices;

(iv) this mixing of sign cannot happen inside the universally tight standard neighborhoodwith slope(Γ) = −∞.

This contradicts s > 0. So tb(K(P,q)) = Pq.

We now show that any N(P,q) can be thickened to a standard neighborhood of L(P,q)

with t(L(P,q)) = 0. So suppose that N(P,q) has convex boundary with slope(Γ∂N(P,q)) = s, as

measured in the C′ framing, where −∞′ < s < 0. Construct R as in Step 1 above, and lookat the convex annulus A, which in this case may not be standard convex. If all dividingcurves on A are boundary parallel arcs, then N(P,q) can be thickened to have boundaryslope ∞′. On the other hand, if there are nonseparating dividing curves on A after goingthrough bypasses, then the resulting T2 will have negative boundary slope in the C′′ framing(since s < 0 in the C′ framing), and we can thicken Nr to obtain a convex torus outside ofR on the T2-side with slope q/P in the C′ framing, since q/P < 0 and thickening can occur.Then using the Imbalance Principle we can thicken N(P,q) to have boundary slope ∞′.

It remains to show that we can achieve just two dividing curves for this N(P,q). Notethat N(P,q) is contained in a thickened torus R representing K with ∂R = T2 − T1 andwhere the dividing curves on Ti have slope q/P . The key now is that since q/P < 0, thereis twisting on both sides of R. We can thus reduce the number of dividing curves on N(P,q)

by either finding bypasses in R\N(P,q) or by finding bypasses along T1 or T2 that can beextended into R, as in the proofs of Claims 4.1 and 4.3 in [17]. �


4.1.4 Definition of a χ-sequence of solid tori

We have shown how under the right conditions, we can establish that certain cablings satisfythe UTP. We will now begin to investigate how cabling can lead to particular candidatesfor non-thickenable solid tori.

Let K be a knot type; we define a sequence of solid tori representing K, which may ormay not exist for a given knot type.

Specifically, define a χ-sequence of solid tori for K to be a sequence of solid tori Nk,with k a nonnegative integer and such that the following hold:

1. N0 is a standard neighborhood of a Legendrian knot at tb(K).

2. For k > 0, Nk is a universally tight solid torus representing K with intersectionboundary slope −(k + 1)/χ(K).

For a given χ-sequence, we will also define the values nk = gcd((k + 1),−χ(K)) fork > 0, and we define n0 = 2.

Note that for a given k, there can actually be multiple non-contactomorphic solid torirepresenting Nk, since there can be multiple universally tight solid tori with the sameboundary conditions.

We will usually just refer to “the Nk” to indicate a χ-sequence.

4.1.5 Definition of a χ-candidate knot type

A χ-candidate knot is essentially a knot type K for which all potential non-thickenable solidtori are members of its χ-sequence. Specifically, a χ-candidate knot is a knot K for whichthe following hold:

1. w(K) = tb(K) > 0.

2. Any solid torus N representing K thickens to an Nk for some k.

3. If N is a solid torus (representing K) which fails to thicken, then N has the sameboundary slope as some Nk, as well as at least 2nk dividing curves.

Note that a χ-candidate knot does not necessarily fail the UTP, as every N may thickento N0. However, if it does fail the UTP, then it does so due to the existence of non-trivialnon-thickenables whose intersection boundary slopes are −(k + 1)/χ(K) < w(K) = tb(K).

Note that a χ-candidate knot K always has lw(K) ≥ 0, since slope(Γ∂Nk) > 0 for all k.Thus, by Theorem 4.1.3, negative cablings of χ-candidate knots satisfy the UTP.

This definition may seem technical, but we will see in the next section that positivetorus knots are χ-candidate knots; by the following Theorem 4.1.4, then, all iterated torusknots Kr where Pi > 0 will be χ-candidate knots.


4.1.6 Positive cabling preserves χ-candidate knots

In this section we prove the following structure theorem.

Theorem 4.1.4 If K is a χ-candidate knot, then K(P,q) is a χ-candidate knot, providedP > 0.

We break the argument into two cases, Case I being where P/q > w(K), and Case IIbeing where w(K) > P/q > 0.

Case I: P/q > w(K)

We first prove part of item 1 of the definition of χ-candidate knots.

Lemma 4.1.5 If P/q > w(K) for K a χ-candidate knot, then tb(K(P,q)) = Pq − (P −qw(K)) > 0.

Proof. The proof is similar to that of Lemma 3.3 in [17]. We first claim that t(K(P,q)) < 0.If not, there exists a Legendrian L(P,q) with t(L(P,q)) = 0 and a solid torus N with L(P,q)

as a Legendrian divide. But then we would have a boundary slope of P/q > w(K) in the Cframing, which cannot occur.

So since t(K(P,q)) < 0, any Legendrian L(P,q) must be a ruling on a convex ∂N withslope s. But then the argument in the second paragraph of Lemma 3.3 in [17] shows thattb(K(P,q)) = Pq − (P − qw(K)) is achieved by a Legendrian ruling where s = 1/w(K).

Finally, note that Pq − (P − qw(K)) = P (q − 1) + qw(K) > 0. �

We now prove items 2 and 3 of the definition of χ-candidate knots.

Lemma 4.1.6 Let N(P,q) be a solid torus representing K(P,q), where P/q > w(K) for K a

χ-candidate knot. Then N(P,q) can be thickened to an Nk′

(P,q) for some nonnegative integer

k′. Moreover, if N(P,q) fails to thicken, then it has the same boundary slope as some Nk′

(P,q),

as well as at least 2nk′

(P,q) dividing curves.

Proof. Let N(P,q) be a solid torus representing K(P,q). Let L be a Legendrian representativeof K in S3\N(P,q) and such that we can join ∂N(L) to ∂N(P,q) by a convex annulus A(P,q)

whose boundaries are (P, q) and ∞′ rulings on ∂N(L) and ∂N(P,q), respectively. Thentopologically isotop L in the complement of N(P,q) so that it maximizes tb over all suchisotopies; this will induce an ambient topological isotopy of A(P,q), where we still can assumeA(P,q) is convex. In the C framing we will have slope(Γ∂N(L)) = 1/m where m ∈ Z. Nowif m = w(K), then there will be no bypasses on the ∂N(L)-edge of A(P,q), since the (P, q)ruling would be at maximal twisting. On the other hand, if m < w(K), then there willstill be no bypasses on the ∂N(L)-edge of A(P,q), since such a bypass would induce adestabilization of L, thus increasing its tb by one by the twist number lemma (Lemma3.1.31). To satisfy the conditions of this lemma, we are using the fact that P/q > w(K).


Figure 4.3: N(P,q) is the larger solid torus in gray; N(L) is the smaller solid torus in gray.

Furthermore, we can thicken N(P,q) through any bypasses on the ∂N(P,q)-edge, and thusassume A(P,q) is standard convex. See (a) in Figure 4.3.

Now let N := N(P,q) ∪ N(A(P,q)) ∪ N(L). By our inductive hypothesis we can thicken

N to an N with intersection boundary slope −(k + 1)/χ(K), and we can assume that k isminimized for all such thickenings. Then consider a convex annulus A from ∂N(L) to ∂N ,such that A is in the complement of N(P,q) and ∂A consists of (P, q) rulings. See (b) in

Figure 4.3. We will show that A is standard convex. Certainly there are no bypasses onthe ∂N(L)-edge of A; furthermore, any bypasses on the ∂N -edge must pair up via dividingcurves on ∂N and cancel each other out as in part (a) of Figure 4.4, for otherwise a bypass on∂N(L) would be induced via the annulus A as in part (b) of Figure 4.4. As a consequence,allowing N to thin inward through such bypasses does not change the boundary slope, butjust reduces the number of dividing curves. But then inductively we can thicken this newN to a smaller k-value, contradicting the minimality of k. Thus A is standard convex.

Now four annuli compose the boundary of a solid torus N(P,q) containing N(P,q): the two

sides of a thickened A; ∂N\∂A; and ∂N(L)\∂A. We can compute the intersection boundaryslope of this solid torus. To this end, recall that slope(Γ∂N(L)) = 1/m. To determine m

we note that the geometric intersection of (P, q) with Γ on ∂N and ∂N(L) must be equal,yielding the equality

P − mq = Pk + P + qχ(K) (4.3)

This gives

−m = Pk

q+ χ(K) (4.4)


Figure 4.4: Part (a) shows bypasses that cancel each other out after edge-rounding. Part (b) shows

a bypass induced on ∂N(L) via A.

We define the integer k′ := k/q. We now choose (P ′, q′) to be a curve on these twotori such that Pq′ − P ′q = 1, we change coordinates to C′′ via the map ((P, q), (P ′, q′)) 7→((0, 1), (−1, 0)). Under this map we obtain

slope(Γ∂N

) =P ′(k + 1) − q′(−χ(K))

P (k + 1) + qχ(K)(4.5)

slope(Γ∂N(L)) =q′P k

q+ P ′ + q′χ(K)

P (k + 1) + qχ(K)(4.6)

We then obtain in the C′ framing, after edge-rounding, that the intersection boundaryslope of N(P,q) is

slope(Γ∂Nr

) =P ′(k + 1) − q′(−χ(K))

P (k + 1) + qχ(K)

−q′P k

q+ P ′ + q′χ(K)

P (k + 1) + qχ(K)

−1

P (k + 1) + qχ(K)

= −k′ + 1

P (k′q + 1) + qχ(K)(4.7)

By then changing coordinates from the C′ framing to the C framing we obtain −(k′ +1)/χ(K(P,q)).

We now prove that the contact structure on this Nk′

(P,q) is universally tight. To see this

note that Nk′

(P,q) is embedded inside a Nk with a universally tight contact structure. Now

there is a q-fold cover of Nk that maps q lifts Nk′

(P,q) to Nk′

(P,q), the lifts themselves each

being an S1 × D2 . This cover in turn has a universal cover R × D2 that contains q copies


of a universal cover R × D2 of Nk′

(P,q). Since the universal cover of Nk has a tight contact

structure, a tight contact structure is thus induced on the universal cover of Nk′

(P,q).

We have shown that any N(P,q) representing K(P,q) can be thickened to one of the Nk′

(P,q),

and if N(P,q) fails to thicken, then it has the same boundary slope as some Nk′

(P,q). We nowshow that if N(P,q) fails to thicken, and if it has the minimum number of dividing curves

over all such N(P,q) which fail to thicken and have the same boundary slope as Nk′

(P,q), then

N(P,q) is actually an Nk′

(P,q).

To see this, as above we can choose a Legendrian L that maximizes tb in the complementof N(P,q) and such that we can join ∂N(L) to ∂N(P,q) by a convex annulus A(P,q) whoseboundaries are (P, q) and ∞′ rulings on ∂N(L) and ∂N(P,q), respectively. Again we haveno bypasses on the ∂N(L)-edge, and in this case we have no bypasses on the ∂N(P,q)-edgesince N(P,q) fails to thicken and is at minimum number of dividing curves.

As above, let N := N(P,q) ∪ N(A(P,q)) ∪ N(L). We claim this N fails to thicken. To

see this, take a convex annulus A from ∂N(L) to ∂N , such that A is in the complement ofN(P,q) and ∂A consists of (P, q) rulings. We know A is standard convex since the twisting isthe same on both edges and there are no bypasses on the ∂N(L)-edge. A picture is shownin Figure 4.5.

Figure 4.5: Shown is a meridional cross-section of N . The larger gray solid torus represents N(P,q);the smaller gray solid torus is N(L).

Now four annuli compose the boundary of a solid torus containing N(P,q): the two sides

of the thickened A, which we will call A+ and A−; ∂N\∂A, which we will call AN ; and∂N(L)\∂A, which we will call AL. Any thickening of N will induce a thickening of N(P,q)

to N(P,q) via these four annuli.

Suppose, for contradiction, that N thickens outward so that slope(Γ∂N ) changes. Notethat during the thickening, AL stays fixed. We examine the rest of the annuli by breakinginto two cases.


Case 1: After thickening, suppose A is still standard convex; that means both A+

and A− are standard convex. Since we can assume that after thickening AN is still stan-dard convex, this means that in order for slope(Γ∂N ) to change, the holonomy of ΓAN

must have changed. But this will result in a change in slope(Γ∂N(P,q)), since AL stays

fixed and any change in holonomy of ΓA+

and ΓA−

cancels each other out and does not

affect slope(Γ∂N(P,q)). Thus we would have a slope-changing thickening of N(P,q), which by

hypothesis cannot occur.Case 2: After thickening, suppose A is no longer standard convex. Now note that there

are no bypasses on the ∂N(L)-edge of A; furthermore, any bypass for A+ on the ∂N -edgemust be cancelled out by a corresponding bypass for A− on the ∂N -edge as in part (a)of Figure 4.4, so as not to induce a bypass on the ∂N(L)-edge as in part (b) of the samefigure. But then again, in order for slope(Γ∂N(P,q)

) to remain constant, the holonomy ofΓAN

must remain constant, and thus slope(Γ∂N ) must also have remained constant, withjust an increase in the number of dividing curves.

This proves the claim that N does not thicken, and we therefore know that its boundaryslope is −(k + 1)/χ(K). Furthermore, we know the number of dividing curves is 2n wheren ≥ nk. Suppose, for contradiction, that n > nk. Then we know we can thicken N to anNk, and if we take a convex annulus from ∂N to ∂Nk whose boundaries are (P, q) rulings,by the Imbalance Principle there must be bypasses on the ∂N -edge. But these would inducebypasses off of ∞′ rulings on N(P,q), which by hypothesis cannot exist. Thus n = nk, andby a calculation as above we obtain that the intersection boundary slope of N(P,q) must be−(k′ + 1)/χ(K(P,q)) for the integer k′ = k/q. �

We complete this case by showing that w(K(P,q)) = tb(K(P,q)).

Lemma 4.1.7 If P/q > w(K) for K a χ-candidate knot, then w(K(P,q)) = tb(K(P,q)).

Proof. We show that 1/tb(K(P,q)) < −(k′ + 1)/χ(K(P,q)) for any candidate Nk′

(P,q) where

k′ > 0. As a consequence, since any N(P,q) thickens to some Nk′

(P,q) (including k′ = 0), we

have, to prevent overtwisting, that w(K(P,q)) = tb(K(P,q)).

We have that 1/tb(K(P,q)) < −(k′ + 1)/χ(K(P,q)) holds if and only if

−χ(K(P,q)) < (k′ + 1)(tb(K(P,q)) (4.8)

Inductively we know that 1/tb(K) < −(k + 1)/χ(K) where k = k′q. This implies that

−χ(K) < (k′q + 1)w(K) (4.9)

We can now prove inequality 4.8; we begin with −χ(K(P,q)). We have:

−χ(K(P,q)) = −qχ(K) − P + Pq

< q(k′q + 1)w(K) − P + Pq


= (k′ + 1)(Pq − (P − qw(K)) − k′(q − 1)(P − qw(K))

< (k′ + 1)(Pq − (P − qw(K))

= (k′ + 1)tb(K(P,q)) (4.10)

�

Case II: w(K) > P/q > 0

We first prove item 1 of the definition of χ-candidate knots.

Lemma 4.1.8 If w(K) > P/q > 0 for K a χ-candidate knot, then tb(K(P,q)) = w(K(P,q)) =Pq > 0.

Proof. The proof is similar to that of Theorem 4.1.3. However, since P/q is not necessarilyless than the lower width, we will need to make appropriate modifications to the proof.

We first examine representatives of K(P,q) at tb. Since there exists a convex torusrepresenting K with Legendrian divides that are (P, q) cablings (inside of the solid torusrepresenting K with slope(Γ) = 1/w(K)) we know that tb(K(P,q)) ≥ Pq. To show that

tb(K(P,q)) = Pq, we show that t(K(P,q)) = 0 by showing that the contact width w(K(P,q), C′) =

0, since this will yield tb(K(P,q)) ≤ w(K(P,q)) = Pq. So suppose, for contradiction, that someN(P,q) has convex boundary with slope(Γ∂N(P,q)

) = s > 0, as measured in the C′ framing,and two dividing curves. After shrinking N(P,q) if necessary, we may assume that s is a largepositive integer. Then let A be a convex annulus from ∂N(P,q) to itself having boundarycurves with slope ∞′. Taking a neighborhood of N(P,q) ∪A yields a thickened torus R withboundary tori T1 and T2, arranged so that T1 is inside the solid torus N representing Kbounded by T2.

Since w(K) ∈ Z, there exists an oriented diffeomorphism of ∂N(K) that takes (w(K), 1)to (0, 1) and fixes (1, 0). Under this change of coordinates, (P, q) becomes a negative cabling.We can then assume in the argument that follows, without loss of generality, that w(K) = 0and P < 0.

Now there are no boundary parallel dividing curves on A, for otherwise, we could passthrough the bypass and increase s to ∞′, yielding excessive twisting inside N(P,q). HenceA is in standard form, and consists of two parallel nonseparating arcs. We now choosea new framing C′′ for N where (P, q) 7→ (0, 1); then choose (P ′′, q′′) 7→ (1, 0) so thatP ′′q − q′′P = 1 and such that slope(ΓT1) = −s and slope(ΓT2) = 1. As mentioned in thethird paragraph of the proof of Theorem 1.2 in [17], this is possible since ΓT1 is obtainedfrom ΓT2 by s + 1 right-handed Dehn twists. Then note that in the C′ framing, we havethat q/P > slope(ΓT2) = (q′′ + q)/(P ′′ +P ) > q′′/P ′′, and q/P and q′′/P ′′ are connected byan arc in the Farey tessellation of the hyperbolic disc. Thus, since −∞ is connected by anarc to 0/1 in the Farey tessellation, we must have that (q′′ + q)/(P ′′ + p) < 0. Thus we canthicken N to one of the Nk. Then, just as in Claim 4.2 in [17], we have the following:

(i) inside R there exists a convex torus parallel to Ti with slope q/P ;


(ii) R can thus be decomposed into two layered basic slices;

(iii) the tight contact structure on R must have mixing of sign in the Poincare duals ofthe relative half-Euler classes for the layered basic slices;

(iv) this mixing of sign cannot happen inside the universally tight Nk.

This contradicts s > 0. So tb(K(P,q)) = Pq. �

We now prove items 2 and 3 of the definition of a χ-candidate knot.

Lemma 4.1.9 If w(K) > P/q > 0 for K a χ-candidate knot, let N(P,q) be a solid torus

representing K(P,q). Then N(P,q) can be thickened to an Nk′

(P,q) for some nonnegative integer

k′. Moreover, if N(P,q) fails to thicken, then it has the same boundary slope as some Nk′

(P,q),

as well as at least 2nk′

(P,q) dividing curves.

Proof. We begin as we did in case I. If N(P,q) is a solid torus representing K(P,q), as beforechoose L in S3\N(P,q) such that ∂N(L) is joined to ∂N(P,q) by an annulus A(P,q), and withtb(L) maximized over topological isotopies in the space S3\N(P,q). Again refer to Figure4.3.

Now suppose slope(Γ∂N(L)) = 1/m where 1/m < q/P . Then inside N(L) is an N withboundary slope q/P . But then we can extend A(P,q) to an annulus that has no twistingon one edge, and we can thus thicken N(P,q) so it has boundary slope ∞′. Moreover, sincethere is twisting inside N(L), we can assure there are two dividing curves on the thickenedN(P,q). So this situation yields no nontrivial solid tori N(P,q) which fail to thicken.

Alternatively, suppose 1/m is on the arc of the boundary of the Farey tessellation of thehyperbolic disc that goes counterclockwise from q/P to 0. Furthermore, for the momentsuppose 1/(m+1) is also on this arc. Then we can use the twist number lemma to concludethat there are no bypasses on the ∂N(L)-edge of A(P,q), and so we can thicken N(P,q) throughbypasses so that A(P,q) is standard convex. Then the calculation of the boundary slope goes

through as above, and we conclude that N(P,q) thickens to some universally tight Nk′

(P,q).

The Nk that is used for this will have q/P < −(k + 1)/χ(K).For the remaining case, suppose 1/m > q/P and m is the greatest nonnegative integer

satisfying this inequality (so we could have m = 0). We use an oriented diffeomorphism thatsends (m + 1, 1) to (−1, 1) and 0 to 0. Under this map, we will refer to the image of (P, q)as (p′′, q). Thus −1/2 > q/p′′ > −1. Again look at the ∂N(L)-edge of A(P,q). We claimthat this edge has no bypasses. So, for contradiction, suppose it does. Then we can thickenN(L) to a solid torus where the (efficient) geometric intersection of (p′′, q) with dividingcurves is less than p′′+2q. Suppose the slope of this new solid torus is −λ/µ < −1/2, whereλ > 1 since m is minimized in the complement of N(P,q).

We do some calculations. Note first that if 2/µ > 1, then 2 > µ, which means 1 ≥ µ,which implies −1 ≥ −1/µ > −λ/µ, which cannot happen, again since m is minimized inthe complement of N(P,q). Thus we must have 2/µ ≤ 1. But then the geometric intersection

4.2. POSITIVE TORUS KNOTS χ-FAIL THE UTP 59

of (p′′, q) with (−µ, λ) is λp′′ + µq > (µ/2)p′′ + µq ≥ 2/µ[(µ/2)p′′ + µq] = p′′ + 2q. This is acontradiction.

Thus there are no bypasses on the ∂N(L)-edge of A(P,q), and we can thicken N(P,q)

through any bypasses so that A(P,q) is standard convex. The calculations that show N(P,q)

thickens to Nk′

(P,q) go through as in the above lemma, and as in that lemma, the Nk′

(P,q) areuniversally tight.

This shows that any N(P,q) representing K(P,q) can be thickened to one of the Nk′

(P,q),

and if N(P,q) fails to thicken, then it has the same boundary slope as some Nk′

(P,q). We nowshow that if N(P,q) fails to thicken, and if it has the minimum number of dividing curves

over all such N(P,q) which fail to thicken and have the same boundary slope as Nk′

(P,q), then

N(P,q) is actually an Nk′

(P,q).To see this, as above we can choose a Legendrian L that maximizes tb in the complement

of N(P,q) and such that we can join ∂N(L) to ∂N(P,q) by a convex annulus A(P,q) whoseboundaries are (P, q) and ∞′ rulings on ∂N(L) and ∂N(P,q), respectively. Now since N(P,q)

fails to thicken, we can assume that q/P < 1/m and that there are no bypasses on the∂N(L)-edge, and in this case we have no bypasses on the ∂N(P,q)-edge since N(P,q) fails tothicken and is at minimum number of dividing curves.

As above, let N := N(P,q) ∪ N(A(P,q)) ∪ N(L). We claim this N fails to thicken – theproof proceeds identically as in the above lemma, as does the proof that N(P,q) is in fact an

Nk′

(P,q). �

This concludes the proof of Theorem 4.1.4.

4.2 Positive torus knots χ-fail the UTP

4.2.1 Definition of χ-failure of the UTP

A knot type K χ-fails the UTP if K is a χ-candidate knot type such that a C-tail of itsχ-sequence exists and represents actual non-thickenable solid tori.

Note that if K χ-fails the UTP, then lw(K) = 0.In the rest of this section we will show that positive torus knots χ-fail the UTP.

4.2.2 Positive torus knots are χ-candidate knots

We will show that any solid torus N representing K = (P, q) can be thickened to an Nk forsome nonnegative integer k, and that any solid torus with the same boundary slope as Nk

which fails to thicken must have at least 2nk dividing curves. Another way of saying thisis that every solid torus N is contained in some Nk, and that if N fails to thicken, thenboundary slopes do not change in passing to the Nk ⊃ N , although the number of dividingcurves may decrease.

Lemma 4.2.1 Let N be a solid torus with core K = (P, q) where P, q > 1 and co-prime.Then N can be thickened to an Nk for some nonnegative integer k. Moreover, if N fails to


thicken, then it has the same boundary slope as some Nk, as well as at least 2nk dividingcurves.

Proof. We first construct the setting. Let T be a torus which bounds solid tori V1 and V2

on both sides in S3, and which contains a (P, q) torus knot K. We will think of T = ∂V1

and T = −∂V2. Let Fi be the core unknots for Vi. We know tb(K) = Pq − p − q = −χ(K)(see [18]); measured with respect to the coordinate system C′, for either i, t(K) = −P − q.

Now let Li, i = 1, 2, be a Legendrian representative of Fi with tb = −mi, wheremi > 0 (recall that tb(unknot) = −1). If N(Li) is a regular neighborhood of Li, thenslope(Γ∂N(Li)) = −1/mi with respect to CFi

.Consider an oriented basis ((P, q), (P ′, q′)) for T , where Pq′ − qP ′ = 1; we map this

to ((0, 1), (−1, 0)) in a new framing C′′. This corresponds to the map Φ1 =

(q −Pq′ −P ′

).

Then Φ1 maps (−m1, 1) 7→ (−qm1 −P,−q′m1 −P ′). Since we are only interested in slopes,we write this as (qm1 + P, q′m1 + P ′).

Similarly, we change from CF2 to C′′. The only thing we need to know here is that(−m2, 1) maps to (Pm2 + q, P ′m2 + q′).

This concludes the construction of the setting; we can now prove the lemma. Let N bea solid torus representing K. Let Li be Legendrian representatives of Fi which maximizetb(Li) in the complement of N , subject to the condition that L1 ⊔L2 is isotopic to F1 ⊔ F2

in the complement of N .Now suppose qm1 + P 6= Pm2 + q. This would mean that the twisting of Legendrian

ruling representatives of K on ∂N(L1) and ∂N(L2) would be unequal. Then we could applythe Imbalance Principle to a convex annulus A in S3\N between ∂N(L1) and ∂N(L2) tofind a bypass along one of the ∂N(Li). This bypass in turn gives rise to a thickening ofN(Li), allowing the increase of tb(Li) by one by the twist number lemma. Hence, eventuallywe arrive at qm1 + P = Pm2 + q and a standard convex annulus A.

Since mi > 0, the smallest solution to qm1+P = Pm2+q is m1 = m2 = 1. All the otherpositive integer solutions are therefore obtained by taking m1 = Pk + 1 and m2 = qk + 1with k a nonnegative integer. We can then compute the intersection boundary slope of thedividing curves on ∂(N(L1)∪N(L2)∪A), measured with respect to C′, after edge-rounding.This will be the intersection boundary slope for N ⊃ N . We have:

−q′(Pk + 1) + P ′

Pqk + P + q+

P ′(qk + 1) + q′

Pqk + P + q−

1

Pqk + P + q= −

k + 1

Pqk + P + q(4.11)

After changing from C′ to C coordinates, these slopes become −(k +1)/χ(K) as desired.This shows that any N thickens to some Nk, and if N fails to thicken, then it has the sameboundary slope as some Nk. Suppose, for contradiction, that N fails to thicken and has 2ndividing curves, where n < nk. Then using the construction above we know that outside ofN in S3 are neighborhoods of the two Legendrian unknots Li with K rulings that intersectthe dividing set on ∂N(Li) exactly 2(Pqk+P +q) number of times. However, since n < nk,the ∞′ rulings on N intersect the dividing set less than 2(Pqk + P + q) number of times.


Thus by the Imbalance Principle there exists bypasses off of the K rulings on the ∂N(Li),and so the Li can destabilize in the complement of N to smaller k-value, allowing for aslope-changing thickening of N . This is a contradiction. �

Note the following inequality, which, among other things, shows that the boundaryslopes of solid tori representing K that may fail to thicken are contained in the interval[−1/(P + q),−1/(Pq)).

−1

P + q< −

2

Pq + P + q< −

3

2Pq + P + q< · · · < −

k + 1

Pqk + P + q< · · · < −

1

Pq(4.12)

We have the following lemma:

Lemma 4.2.2 w(K) = tb(K) = Pq − P − q

Proof. In the C′ framing, we need to show that w(K, C′) = t(K) = −P − q. Usingthe inequality above, it suffices to show that any solid torus N representing K can bethickened to a solid torus with boundary slope −(k+1)/(Pqk+P +q) for some nonnegativeinteger k, for then to prevent overtwisting it would have to be the case that slope(Γ∂N ) ∈[−1/(P + q), 0). But by the above lemma this is true. �

To conclude this section, we have the following lemma that shows that the potentialnon-thickenables have universally tight contact structures.

Lemma 4.2.3 If Nk fails to thicken, then its contact structure is universally tight; more-over, for k > 0, a convex meridian disc contains bypasses that all bound half-discs of thesame sign. Also, a Legendrian ruling preferred longitude on ∂Nk has rotation number zerofor k > 0.

Proof. The lemma is immediately true for k = 0, so we may assume that k > 0. To fixnotation, let L1 be the Legendrian unknot with tb = −(Pk + 1) and let L2 be the unknotwith tb = −(qk + 1). Then N(L1) thickens outward to S3\N(L2); we denote T1 := ∂N(L1)and T2 := ∂(S3\N(L2)).

We then examine a horizontal convex annulus AH in the space (S3\N(L2))\N(L1),bounded by meridian rulings on the Ti. This horizontal convex annulus AH has two dividingcurves that connect its two boundary components; the other qk bypasses have endpointson T2. By Propositon 3.1.28, we may assume that all of these bypasses are boundarycompressible, meaning there are no nested bypasses. The two dividing curves connectingthe two boundary components of AH thus divide AH into two discs, one containing allbypasses of positive sign, the other disc containing all negative bypasses. We will show thatin fact all bypasses on AH must be of the same sign.

To this end, note that, by the construction of the potential non-thickenable Nk, AH

is composed of convex meridian discs D1, ..., Dq for Nk, banded together by q number ofI-invariant bands; see Figure 4.6 where the I-invariant bands are indicated by hatching.


As a consequence, all of the dividing curves for AH are confined to the discs D1, ..., Dq,as no dividing curves pass through the I-invariant bands. So suppose, for contradiction,that the qk bypasses on AH have mixed sign, meaning some are negative and some arepositive. Since each of the q meridian discs is a convex meridian disc for Nk, then by theclassification of tight contact structures on solid tori we know that if one of the discs hasa negative bypass, then all of them must; the same is true for positive bypasses. But sincethe negative (positive) bypasses on AH are grouped in succession and bounded by the twonon-separating dividing curves, and since we may assume q ≥ 3, this forces one of the discsto inherit only negative (positive) bypasses, contradicting the fact that it is supposed toalso have positive (negative) bypasses. Thus all of the bypasses on AH must be of the samesign, as must be all of the bypasses on a convex meridian disc for Nk. As a result thecontact structure for Nk is universally tight.

Figure 4.6: Shown is the horizontal annulus AH ; it is composed of discs D1, ...Dq banded togetherby I-invariant bands as indicated by the hatching. By symmetry, all of the bypasses must be of thesame sign.

We can now calculate the rotation number for the (P, q) ruling on N(L1). Since(S3\N(L2))\N(L1) is universally tight, one can show that if Σ1 is a convex Seifert sur-face for the longitude on ∂N(L1), we must have r(∂Σ1) = ±(Pk). By Lemma 3.1.14, wehave that the (P, q) ruling on ∂N(L1) has rotation number equal to

r((P, q)) = Pr(∂D1) + qr(∂Σ1) (4.13)

This yields r((P, q)) = ±(Pqk). We now let D be a meridian disc for Nk and Σbe a Seifert surface for the preferred longitude on ∂Nk. We know r(∂D) = ±k, andwe know that the (P, q) torus knot, which is ∞′ on ∂Nk, is actually a (Pq, 1) knot on∂Nk

1 in the preferred framing. So using a similar equation from above, we obtain thatr((Pq, 1)) = ±(Pqk) = ±(Pqk) + qr(∂Σ). Thus r(∂Σ) = 0. �


We note that there are two universally tight contact structures, diffeomorphic by −id,which satisfy the conditions set by the above lemma.

4.2.3 The χ-sequence exists with non-thickenable terms

We know that if Nk fails to thicken, its complement Mk := S3\Nk must be contactomorphicto the manifold obtained by taking a neighborhood of a Hopf link N(L1) ⊔ N(L2) and astandard convex annulus A joining the two neighborhoods of the Hopf link, where A hasboundary components that are Legendrian ruling representatives of K = (P, q). Moreover,we know that the two components of the Hopf link must have tb values equal to −(Pk + 1)and −(qk + 1), respectively, for k ≥ 0.

Lemma 4.2.4 The standard tight contact structure on S3 splits into a universally tightcontact structure on Nk and Mk.

Proof. The idea is to build S3. To begin, choose one of the above two universally tightcandidates for Nk. We then claim we can join Nk to itself by a standard convex annulus A′

with boundary ∞′ rulings so that R := Nk ∪N(A′) is a (universally tight) thickened toruswith boundary T2 −T1 having associated boundary slopes of −(qk + 1)/1 and −1/(Pk + 1)and two dividing curves. One way to see this is that we can think of ∂Nk as being composedof four annuli, one from T2\A

′, one from −T1\A′, and two from ∂A′ × [−ǫ, ǫ]. Since we are

constructing the thickened torus, with a suitable choice of holonomy of A′, we can assurethat the dividing curves on −T1 have only one longitude, and two components. Since weknow the twisting of ∞′ on Nk

1 is equal to −(Pqk + P + q), a calculation shows that thedividing curves on −T1 must have slope −1/(Pk + 1). But then the slopes of the dividingcurves on −T1 and ∂Nk are determined, making the slope of dividing curves on T2 equalto −(qk + 1)/1 based on equation 4.11.

Now as in the proof of Lemma 5.2 in [17], the contact structure on Nk ∪ N(A′) canbe isotoped to be transverse to the fibers of Nk ∪ N(A′), which are parallel copies of K1,while preserving the dividing set on ∂(Nk ∪N(A′)). Such a horizontal contact structure isuniversally tight.

We then use the classification of tight contact structures on S3, solid tori, and thickenedtori to conclude that any tight contact structure on R = T 2×[1, 2] with boundary conditionsbeing tori with two dividing curves and slopes −(qk +1)/1 and −1/(Pk +1) glues togetherwith standard neighborhoods of unknots with those boundary slopes to give the tight contactstructure on S3. �

We now show that these Nk with complements Mk fail to thicken. This lemma is jointwork with John Etnyre and Bulent Tosun.

Lemma 4.2.5 The Nk with complement Mk fail to thicken.

Proof. By inequality 4.12, it suffices to show that Nk does not thicken to any Nk′

fork′ < k. So to this end, observe that the (P, q) positive torus knot is a fibered knot over


S1 with fiber a Seifert surface Σ of genus g = ((P − 1)(q − 1))/2 (see [38]). Moreover, the

monodromy is periodic with period Pq. Thus, Mk has a Pq-fold cover Mk ∼= S1×Σ. If onethinks of Mk as Σ× [0, 1] modulo the relation (x, 0) ∼ (φ(x), 1) for monodromy φ, then one

can view Mk as Pq copies of Σ × [0, 1] cyclically identified via the same monodromy. Nownote that downstairs in Mk, ∞′ intersects any given Seifert surface Pq times efficiently. Itis therefore evident that we can view Mk as a Seifert fibered space with base space Σ andtwo singular fibers (the components of the Hopf link). The regular fibers are topologicalcopies of ∞′, which itself is a Legendrian ruling on ∂Nk

1 with twisting −(Pqk + P + q). Infact, the regular fibers can be assumed to be Legendrian isotopic to the ∂Nk-fibers exceptfor small neighborhoods around the singular fibers.

We claim the pullback of the tight contact structure to Mk admits an isotopy where theS1 fibers are all Legendrian and have twisting number −(Pqk + P + q) with respect to the

product framing. This isotopy can be accomplished because in Mk, the lifts of the singularfibers have tight neighborhoods with convex boundary tori which have dividing curves withone longitude and where ∞′ has twisting −(Pqk + P + q). Thus these neighborhoods ofthe lifts of the singular fibers are in fact standard neighborhoods of a Legendrian fiber withtwisting −(Pqk + P + q); the contact structure can then be isotoped so that every fiberinside these neighborhoods is Legendrian with twisting −(Pqk + P + q).

So, if Nk can be thickened to Nk′

, then there exists a Legendrian curve topologicallyisotopic to the regular fiber of the Seifert fibered space Mk with twisting number greaterthan −(Pqk + P + q), measured with respect to the Seifert fibration. Pulling back to the

Pq-fold cover Mk, we have a Legendrian knot which is topologically isotopic to a fiber buthas twisting greater than −(Pqk + P + q). Call this Legendrian knot with greater twistingγ. We will obtain a contradiction, thus proving that Nk cannot be thickened to Nk′

.Since Σ is a punctured surface of genus g, we can cut Σ along 2g disjoint arcs αi, all

with endpoints on ∂Σ, that yield a polygon P . Thus we have a solid torus S1×P embeddedin Mk. We first need to calculate slope(Γ∂(S1×P )) as measured in the product framing. Todo so, note that a longitude for this torus intersects Γ 2(Pqk+P +q) times, and a meridianfor this torus is composed of 2 copies each of the 2g arcs αi, as well as 4g arcs βi from ∂Σ.Now since ∂Σ is a preferred longitude downstairs in Mk

1 , we know that Γ intersects theseβi 2(Pq − P − q) = 2(2g − 1) times positively (where here we are measuring parities asusual for the boundary of a solid torus). But then the edge-rounding that results at eachintersection of an S1 × βi with an S1 × αi yields 4g negative intersections with Γ. Thus weobtain after edge-rounding that slope(Γ∂(S1×P )) = −1/(Pqk + P + q).

Now as in Lemma 3.2 in [25], we take Mk = S1 × Σ and pass to a (new) finite coverof the base by tiling enough copies of P together so that γ is contained in a solid torusS1 × (

⋃P ). We claim that the lifted contact structure on this new cover (Mk)′ is tight. To

see this, first note that (Mk)′ has a product structure that is S1 times a punctured surfaceF\ {x1, ..., xm}, and all the S1 fibers have twisting −(Pqk+P+q); thus the contact structure

may be isotoped to be horizontal on (Mk)′. Label the boundary component associated tothe puncture xj as µj . Then each S1 ×µj is a torus, and if we use µj as a meridian and theS1 fibers as longitudes, the dividing curves on S1×µj have negative slope as measured using

4.3. UTP CLASSIFICATION OF ITERATED TORUS KNOTS 65

an orientation pointing into the fibered manifold. It is then evident that at each S1 × µj

we can glue in a tight solid torus, so that we obtain S1 times a closed surface F with ahorizontal contact structure. Thus the contact structure on S1 × F is tight, and hence therestriction of the contact manifold to (Mk)′ is tight as well.

We now shift our perspective back solely to the tight (Mk)′, and simply note that S1 ×(⋃

P ) is a tight solid torus neighborhood of a Legendrian knot of twisting −(Pqk + P + q),and thus by the classification of tight contact structures on solid tori, cannot contain γ.This is our contradiction. �

Corollary 4.2.6 For a positive torus knot K, 0 = lw(K) < w(K) = Pq − P − q; thuspositive torus knots fail the UTP.

Proof. The above lemma shows that lw(K, C′) = −Pq, which after a change of coordinatesyields lw(K) = 0. �

4.3 UTP classification of iterated torus knots

In this section we will provide a complete UTP classification of iterated torus knots, specif-ically identifying what knot types satisfy, or fail, the UTP.

Theorem 4.3.1 Let Kr = ((P1, q1), ..., (Pi, qi), ..., (Pr, qr)) be an iterated torus knot, wherethe Pi’s are measured in the standard preferred framing, and qi > 1 for all i. Then Kr failsthe UTP if and only if Pi > 0 for all i, where 1 ≤ i ≤ r.

To prove this theorem it will suffice to study iterated torus knots Kr for which Pi > 0for all i, and show that they χ-fail the UTP, since this will imply that their lower widthsare zero, and hence any iterated torus knot with a negative iteration satisfies the UTP byTheorems 4.1.1 and 4.1.3.

A corollary to Theorem 4.3.1 and Section 2.4.4 is the following:

Corollary 4.3.2 An iterated torus knot Kr fails the UTP if and only if ξKr∼= ξstd.

4.3.1 The χ-sequence Nkr

Let Kr be an iterated torus knot where Pi > 0 for all i. We will refer to terms in theχ-sequence of Kr as Nk

r , and we note these Nkr will have intersection boundary slopes of

−(k + 1)/(Ark + Br), as measured in the C′ framing. Note that in this notation the Nk

representing torus knots will be labelled as Nk1 .

4.3.2 Any Kr with Pi > 0 for all i χ-fails the UTP

First note that it is immediate, from subsections 4.1.6 and 4.2.2, that any Kr with Pi > 0for all i is a χ-candidate.


Inductively we can assume that Kr χ-fails the UTP, where Pi > 0 for all i; we then showthat Kr+1 χ-fails the UTP provided Pr+1 > 0. Furthermore, we can assume that the Nk

r arenot only universally tight, but have convex meridian discs whose bypasses bound half-discsall of the same sign. Moreover, the preferred longitude on ∂Nk

r has rotation number zerofor k > 0. We then have that any χ-candidate terms Nk′

r+1 must be embedded inside Nkr

which fail to thicken, and such that Nk′

r+1 = Nkr \(N(Lr) ∪ A(Pr+1,qr+1)), where A(Pr+1,qr+1)

is a standard convex annulus. We then prove the following lemma.

Lemma 4.3.3 The candidate Nk′

r+1 exist for k′ ≥ Cr+1, where Cr+1 is some positive integer.

Moreover, these Nk′

r+1 have convex meridian discs whose bypasses bound half-discs all of the

same sign. Also, the preferred longitude on ∂Nk′

r+1 has rotation number zero for k′ > 0.

Proof. To see that a meridian disc for Nk′

r+1 contains bypasses all of the same sign, note that

this is immediate if ∂Nk′

r+1 has two dividing curves. For the case of 2n dividing curves wheren > 1, we argue in a similar fashion to Lemma 4.2.3 for positive torus knots. Specifically,since a meridian disc for Nk

r inductively has bypasses all of the same sign, a horizontalannulus AH with boundary on ∂Nk

r and ∂N(Lr) will have qrk′ bypasses all of the same

sign. Thus, as in the above lemma, a meridian disc for Nk′

r+1 will inherit k′ + 1 bypasses allof the same sign.

To show that the preferred longitude on ∂Nk′

r+1 has rotation number zero, we again usean argument similar to that used for positive torus knots. We call the meridian disc for Nk

r ,Dr, and the Seifert surface for the preferred longitude on ∂Nk

r , Σr. If we then look at the(Pr+1, qr+1) cable on ∂Nk

r , we can calculate its rotation number as

r((Pr+1, qr+1)) = Pr+1r(∂Dr) + qr+1r(∂Σr) = Pr+1(±qr+1k′) (4.14)

But then since this same knot is a (Pr+1qr+1, 1) cable on ∂Nk′

r+1, we have that r((Pr+1, qr+1))= Pr+1qr+1(±k′) + qr+1r(∂Σ), where Σ is a Seifert surface for the preferred longitude on∂Nk′

r+1. This implies that r(∂Σ) = 0.Now we know inductively that there exists a Cr such that if k ≥ Cr, then the Nk

r

exist and fail to thicken. So suppose k/qr+1 ∈ N for some k ≥ Cr. Also assume thatqr+1/Pr+1 < −(k + 1)/χ(Kr); we know such a k exists since −(k + 1)/χ(Kr) → ∞ as kincreases. We will show that Nk′

r+1 exists for k′ := k/qr+1. Then Cr+1 will be the least suchk/qr+1 ∈ N.

We take one of the two universally tight candidate Nk′

r+1, and as in Lemma 4.2.4 weconstruct a universally tight R and glue in an appropriate solid torus neighborhood of aLegendrian knot Lr to obtain a universally tight Nk

r , which then glues into S3 inductively.This shows that Nk′

r+1 exists. �

Lemma 4.3.4 The candidate Nk′

r+1 actually fail to thicken.

Proof. To show that Nk′

r+1 fails to thicken, it suffices to show that Nk′

r+1 does not thicken to

any Nk′′

r+1, where k′′ < k′. Inductively, we can assume Nkr fails to thicken; in particular, the

4.3. UTP CLASSIFICATION OF ITERATED TORUS KNOTS 67

Nk′qr+1r that contains Nk′

r+1 fails to thicken. So let k = k′qr+1. Then define Mkr = S3\Nk

r ,

and define Mk′

r+1 = S3\Nk′

r+1.

Let Σr+1 be a Seifert surface for a preferred longitude on ∂Nk′

r+1; Σr+1 is a puncturedsurface of genus g′. Now there are qr+1 separating simple closed curves on Σr+1 that are infact preferred longitudes for ∂Nk

r , and thus bound Seifert surfaces Σr for the knot Kr. Wewill call the genus of the punctured surface Σr, g. Also we will call Σr+1 ∩ Nk

r := σr+1.

Now we look at finite covers of Mk′

r+1 that are obtained by cutting Mk′

r+1 along Σr+1 and

then cyclically stacking copies of these split-open Mk′

r+1. We first look at a Pr+1qr+1-foldcover obtained in this fashion. If we arrange that downstairs Nk

r has Legendrian rulingsthat are (Pr+1, qr+1) cables, then upstairs in the cover the lift of Nk

r can be fibered byLegendrian fibers all with twisting −(Ar+1k

′ + Br+1). Moreover, the rest of the cover isqr+1 copies of Pr+1-fold covers of Mk

r , each of which is fibered (horizontally) by Σr.

Inductively we can assume that the monodromy for the fibered space Mkr is periodic with

period mr, and also that a resulting mr-fold product cover can be fibered by Legendrianfibers that all have twisting −sr(Ark + Br), where sr is some positive integer.

As a consequence, we can now cyclically stack mr copies of our Pr+1qr+1-fold cover ofMk′

r+1 to obtain Mk′

r+1. This Mk′

r+1 will be a product space S1 × Σr+1. Thus Kr+1 is afibered knot having periodic monodromy, and we claim that the period is some multipleof Pr+1qr+1. This is because the Seifert surface Σr+1 is formed by taking qr+1 copiesof Σr, and Pr+1 copies of a meridian disc Dr for Nr, and banding them together withPr+1qr+1 positive (half-twist) bands. Therefore, to at least cycle through the Dr’s andΣr’s set-wise, the monodromy must be some multiple of Pr+1qr+1. Hence, if we restrict toS1 × σr+1 ⊂ S1 × Σr+1, the space S1 × σr+1 can be fibered by Legendrians all of twisting−sr+1(Ar+1k

′ + Br+1), for some positive integer sr+1, with respect to the product framing.However, at the moment all we know is that the qr+1 copies of S1 × Σr can be fibered bytopological copies of these Legendrian fibers in S1 × σr+1; what we will show is that in factS1 × Σr+1 can be fibered by Legendrian copies of the fibers in S1 × σr+1.

To this end, downstairs let T = ∂Nkr . We may assume that the rulings on T are copies

of ∞′r+1, and the solid torus bounded by T lifts to S1 × σr+1, where all the S1 fibers are

Legendrian isotopic to lifts of ∞′r+1, and have twisting −sr+1(Ar+1k

′ + Br+1) for somepositive integer sr+1. We will call these S1 fibers S1

r+1, and note that they are topologicallyisotopic to the S1 fibers in the product space S1 × Σr+1. We also have that if we think ofMk

r as bounded by T , then Mkr lifts to qr+1 copies of S1 × Σr, where all the S1 fibers are

Legendrian isotopic to lifts of ∞′r, and have twisting −srPr+1(Ark +Br). We will call these

S1 fibers S1r . We will show that in fact, all of Mk′

r+1 can be fibered by Legendrian S1r+1’s.

On the Seifert surface Σr+1, we will label the qr+1 Σr’s as Σjr. Now let αi

r+1 be 2g′

disjoint arcs on Σr+1, each with endpoints on ∂Σr+1, and such that if we cut along theαi

r+1 we obtain a polygon Pr+1. Moreover, arrange it so that of the αir+1, qr+12g of them

leave σr+1 and restrict to arcs on the qr+1 copies of Σr that, when we cut along them, yieldpolygons P j

r .

Now the arcs αir+1 that stay in σr+1 represent an interval’s worth of S1

r+1 fibers oftwisting −sr+1(Ar+1k

′ + Br+1), and hence represent standard convex annuli in the space


Mk′

r+1. The arcs αir+1 that leave σr+1 represent convex annuli that are fibered by S1

r+1’swhen restricted to their intersection with the lift of the solid torus bounded by T . So whatis of interest is a convex annulus Ai with boundary components that both have twisting−sr+1(Ar+1k

′ + Br+1), fibered by topological copies of the S1r+1’s but which is embedded

in one of the qr+1 lifts of Mkr .

So suppose, for contradiction, that there exists a bypass on one of the Ai’s. We lookat what passing through this bypass will do on the lift of T to which Ai is attached; weuse the framing on T that comes from the the lifts of ∞′

r. If we split the S1r × Σj

r thatcontains the Ai along arcs αi

r+1 to obtain S1r × P j

r , and then pass to a finite cover of the

base by tiling copies of S1r × P j

r (similar to what we did in Lemma 4.2.5), we will obtain abypass off of a ruling inside of a tight standard neighborhood of a Legendrian of twisting−srPr+1(Ark+Br), since the dividing curves on the lift of T intersect ∂Σj

r exactly 2(χ(Kr))times. But this bypass inductively cannot exist due to the classification of tight solid toriand the arguments in Lemma 4.3.

Thus we can conclude that all of the annuli Ai are standard convex. As a consequence,if we now use the framing on S1

r × Σjr that comes from the topological lifts of ∞′

r+1, and

again split the S1r ×Σj

r that contains the Ai along arcs αir+1 to obtain S1

r ×P jr , we actually

obtain a tight standard neighborhood of a Legendrian with twisting −sr+1(Ar+1k′ +Br+1),

since we know that the lifted contact structure restricted to S1r × Σj

r is tight. Thus thecontact structure can be isotoped so that all of the topological S1

r+1 fibers in the space

Mk′

r+1 are in fact Legendrian of twisting −sr+1(Ar+1k′ + Br+1). As a consequence, if we

now split Σr+1 along all arcs αir+1 to obtain S1

r+1×Pr+1, then the boundary torus will havea characteristic foliation that matches that of the standard neighborhood of a Legendrianwith twisting −sr+1(Ar+1k

′ +Br+1), since the dividing curves on the lift of ∂Nk′

r+1 intersect

∂Σr+1 exactly 2(χ(Kr+1) times. Thus the argument that Nk′

r+1 fails to thicken proceeds

exactly as in Lemma 4.3 as soon as we can show that the lifted contact structure on Mk′

r+1

is tight. �

Chapter 5

Results: Simple and non-simpleiterated torus knots

In this chapter we first show that in the class of iterated torus knots, failing the UTP isa sufficient condition for the existence of non-simple cablings. In so doing we establish alarge family of transversally non-simple iterated torus knots. We then also establish largefamilies of Legendrian simple iterated torus knots.

5.1 Transversally non-simple iterated torus knots

In this section we prove the following theorem:

Theorem 5.1.1 If Kr is an iterated torus knot that fails the UTP, then it supports infinitelymany transversally non-simple cablings Kr+1 of the form (−χ(Kr), k + 1), where k rangesover an infinite subset of positive integers.

We will prove this theorem by working through a series of lemmas. These lemmaswill first give us information about just a piece of the Legendrian mountain range forKr = ((P1, q1), ..., (Pr, qr)) where Pi > 0 for all i; we will then use this information toobtain enough information about the Legendrian mountain ranges of certain cables Kr+1 toconclude that these cables are transversally non-simple. We will therefore not be completingthe Legendrian or transversal classification of these iterated torus knots.

In our first lemma we establish the structure of just a piece of the Legendrian mountainrange for Kr:

Lemma 5.1.2 Suppose Kr = ((P1, q1), ..., (Pr, qr)) is an iterated torus knot where Pi > 0for all i. Then there exists Legendrian representatives L±

r with tb(L±r ) = 0 and r(L±

r ) =±(Ar − Br); also, L±

r destabilizes.

Proof. The lemma is true for positive torus knots [18], so we inductively assume it istrue for Kr−1. Then look at Legendrian rulings L±

r on standard neighborhoods of the

69

70CHAPTER 5. RESULTS: SIMPLE AND NON-SIMPLE ITERATED TORUS KNOTS

inductive L±

r−1. In the C′ framing the boundary slope of these N(L±

r−1) is −1/Ar−1, and so

a calculation shows that t(L±r ) = −Pr; hence tb(L±

r ) = Ar − Pr.To calculate the rotation number of L±

r , we use the following formula from [17], where Dis a convex meridian disc for N(L±

r−1) and Σ is a Seifert surface for the preferred longitudeon ∂N(L±

r−1):

r(L±r ) = Prr(∂D) + qrr(∂Σ)

= ±qr(Ar−1 − Br−1)

= ±(qrAr−1 + pr − qrBr−1 − pr)

= ±(Pr − Br)

This gives us

sl(T−(L+r )) = (Ar − Pr) + (Pr − Br) = Ar − Br (5.1)

and

sl(T+(L−r )) = (Ar − Pr) − (−(Pr − Br)) = Ar − Br (5.2)

This, along with Lemma 2.4.2, shows us that L+r is on the right-most slope of the

Legendrian mountain range of Kr, and L−r is on the left-most edge. To the former we can

perform positive stabilizations to reach L+r at tb = 0 and r = Ar −Br; to the latter we can

perform negative stabilizations to reach L−r at tb = 0 and r = −(Ar − Br). We know such

stabilizations can be performed since Ar − Pr > 0. �

So suppose Kr is an iterated torus knot that fails the UTP (which is precisely whenPi > 0 for all i). Then we know that for k ≥ Cr there exist non-thickenable solid tori Nk

r

having intersection boundary slopes of −(k+1)/(Ark+Br), where these slopes are measuredin the C′ framing. Switching to the standard C framing, these intersection boundary slopesare (k+1)/(Ar−Br) = −(k+1)/χ(Kr). Now as k → ∞, there are infinitely many values ofk+1 which are prime and greater than Ar−Br. As a consequence, there are infinitely manyNk

r with two dividing curves. Based on this observation, we make the following definition:

Definition 5.1.3 Suppose Kr = ((P1, q1), ..., (Pr, qr)) is an iterated torus knot where Pi >0 for all i. Let Kr+1 be a cabling of Kr with C′ slope −(k +1)/(Ark +Br), where −1/(Ar −1) < −(k + 1)/(Ark + Br) < −1/Ar and there is an Nk

r with two dividing curves that failsto thicken.

So given Kr, there are infinitely many such cabling knot types Kr+1, all of these beingcablings of the form (−χ(Kr), k + 1) as measured in the preferred framing. The followinglemma will then prove Theorem 5.1.1.

Lemma 5.1.4 Kr+1 is a transversally non-simple knot type.

5.1. TRANSVERSALLY NON-SIMPLE ITERATED TORUS KNOTS 71

Proof. We first calculate χ(Kr+1). Using the recursive expression we obtain

χ(Kr+1) = qr+1χ(Kr) − Pr+1qr+1 + Pr+1

= (k + 1)(−Ar + Br) − (Ar − Br)(k + 1) + (Ar − Br)

= (2k + 1)(−Ar + Br)

We now look at the two universally tight non-thickenable Nkr that have representatives

of Kr+1 as Legendrian divides. These Legendrian divides have tb = Ar+1 = qr+1Pr+1 =(k + 1)(Ar − Br). To calculate rotation numbers, we have two possibilities, depending onwhich boundary of the two universally tight Nk

r the Legendrian divides reside. Using theformula from [17], we obtain

r(Kr+1) = qr+1r(∂Σ) + Pr+1r(∂D)

= Pr+1(±(qr+1 − 1))

= ±k(Ar − Br)

We will call the two Legendrian divides corresponding to r = ±k(Ar − Br), L±

r+1 re-spectively. We can calculate the self-linking number for the negative transverse push-off ofL+

r+1 to be sl = (2k + 1)(Ar −Br) = −χ(Kr+1). This shows that L+r+1 is on the right-most

edge of the Legendrian mountain range and is at tb. Similarly, L−

r+1 is on the left-most edge

of the Legendrian mountain range and is at tb.

We now look at solid tori Nr with intersection boundary slope −(k + 1)/(Ark + Br),but which thicken to solid tori with intersection boundary slopes −1/(Ar − 1). Such tori∂Nr are embedded in universally tight basic slices bounded by tori with dividing curves ofslope −1/(Ar − 1) and −1/Ar. Legendrian divides on such Nr have tb = (k + 1)(Ar − Br);to calculate possible rotation numbers for these Legendrian divides, we recall the procedureused in the proof of Theorem 1.5 in [30]. There we used a formula for the rotation numbersfrom [17], where the range of rotation numbers was given by the following (substitutingAr − 1 for n):

r(Lr+1) ∈ {±(pr+1 + (Ar − 1)qr+1 + qr+1r(Lr))|tb(Lr) = Ar − (Ar − 1) = 1} (5.3)

Now from Lemma 5.1.2 we know that there is an Lr with tb(Lr) = 1 and r(Lr) =−(Ar −Br)+1. Plugging this value of the rotation number into the expression above yieldsr(Lr+1) = ±k(Ar −Br). We will call the Legendrian divides having these rotation numbersL±

r+1, respectively. Important for our purposes is that L±

r+1 have the same values of tb andr as L±

r+1.

We focus in, for the sake of argument, on L−

r+1 and L−

r+1, and we show that T−(L−

r+1)

is not transversally isotopic to T−(L−

r+1), despite having the same self-linking number.


Consider first T+(L−

r+1). It is in fact one of the dividing curves on ∂Nkr , and is also at

maximal self-linking number for Kr+1. Similarly, T+(L−

r+1) is one of the dividing curves

on ∂Nr, and is also at maximal self-linking number. Now from [26] we know that Kr+1 isa fibered knot that supports the standard contact structure, since it is an iterated torusknot obtained by cabling positively at each iteration. As a consequence, from [19], we alsoknow that Kr+1 has a unique transversal isotopy class at sl. Hence we know that T+(L−

r+1)

and T+(L−

r+1) are transversally isotopic. Thus there is a transverse isotopy (inducing anambient contact isotopy) that takes these two dividing curves on the two different tori toeach other. Thus we may assume that ∂Nk

r and ∂Nr intersect along one component of thedividing curves; we call this component γ+.

Now suppose, for contradiction, that T−(L−

r+1) is transversally isotopic to T−(L−

r+1).These transverse knots are represented by the other two non-intersecting dividing curveson ∂Nk

r and ∂Nr, respectively, and there is a transverse isotopy taking one to the other.We claim that this transverse isotopy can be performed relative to γ+. To see this, notethat associated to S3\N(γ+) is an open book decomposition of S3, with pages being Seifertsurfaces Σ for the knot γ+. Moreover, the standard contact structure is supported by thisopen book decomposition. Thus the transverse isotopy taking T−(L−

r+1) to T−(L−

r+1) willinduce an ambient isotopy of open book decompositions supporting the standard contactstructure, all with a transversal representative of γ+ on the binding. Since ∂Nk

r is incom-pressible in S3\N(γ+), it is therefore evident that the isotopy taking T−(L−

r+1) to T−(L−

r+1)can be accomplished simply as an isotopy of ∂Nk

r relative to γ+.

Thus we may assume that after a contact isotopy of S3, ∂Nkr and ∂Nr intersect along

their two dividing curves, which we denote as γ+ and γ−, and we observe that there isan isotopy (not necessarily a contact isotopy) of Nk

r to Nr relative to γ+ and γ−. Weclaim that as a result Nr cannot thicken, thus obtaining our contradiction. We do this bynoting that the isotopy of Nk

r to Nr relative to γ+ and γ− may be accomplished by theattachment of successive bypasses. Since these bypasses are attached in the complement ofthe two dividing curves, none of these bypass attachments can change the boundary slope.However, they may increase or decrease the number of dividing curves. Starting withT = ∂Nk

r , we make the following inductive hypothesis, which we will prove is maintainedafter bypass attachments:

1. T is a convex torus which contains γ+ and γ−, and thus has slope −(k+1)/(Ark+Br).

2. T is a boundary-parallel torus in a [0, 1]-invariant T 2 × [0, 1] with slope(ΓT0) =slope(ΓT1) = −(k +1)/(Ark +Br), where the boundary tori have two dividing curves.

3. There is a contact diffeomorphism φ : S3 → S3 which takes T 2 × [0, 1] to a standardI-invariant neighborhood of ∂Nk

r and matches up their complements.

The argument that follows is similar to Lemma 6.8 in [17]. First note that item 1 ispreserved after a bypass attachment, since such a bypass is in the complement of γ+ andγ−, and thus cannot change the slope of the dividing curves. To see that items 2 and 3 are

5.2. LEGENDRIAN CLASSIFICATION OF THE ((2, 3), (1, 2)) KNOT 73

preserved, suppose that T ′ is obtained from T by a single bypass. Since the slope was notchanged, such a (non-trivial) bypass must either increase or decrease the number of dividingcurves by 2. Suppose first that the bypass is attached from the inside, so that T ′ ⊂ N ,where N is the solid torus bounded by T . For convenience, suppose T = T0.5 inside theT 2 × [0, 1] satisfying items 2 and 3 of the inductive hypothesis. Then we form the newT 2× [0.5, 1] by taking the old T 2× [0.5, 1] and adjoining the thickened torus between T andT ′. Now T ′ bounds a solid torus N ′, and, by the classification of tight contact structureson solid tori, we can factor a nonrotative outer layer which is the new T 2 × [0, 0.5].

Alternatively, if T ′ ⊂ (S3\N), then we know that N ′ thickens to an Nkr , and thus there

exists a nonrotative outer layer T 2 × [0.5, 1] for S3\N ′, where T1 has two dividing curves.Thus the proof is done, for after enough bypass attachments we will obtain T = ∂Nr, withNr non-thickenable. But this is a contradiction, since Nr does thicken. �

5.2 Legendrian classification of the ((2, 3), (1, 2)) knot

Let K be the (2, 3) torus knot, and K ′ the (1, 2)-cabling of the (2, 3) torus knot. In thissection, we provide a complete Legendrian classification for K ′. In particular, we prove thefollowing theorem:

Theorem 5.2.1 If K ′ is the (1, 2)-cabling of the (2, 3) torus knot, then K ′ has a Legendrianclassification as in Figure 5.1. This entails the following:

1. There are precisely four maximal Thurston-Bennequin representatives of K ′, whichwe call L± and Lnt

± , and which have tb(L±) = tb(Lnt± ) = 2 and r(L±) = r(Lnt

± ) = ±1.

2. Every Legendrian L representing K is a stabilization of one of the L± or Lnt± .

3. S−(L+) = S+(L−), S−(L−) = S−(Lnt− ), and S+(L+) = S+(Lnt

+ ).

4. Sn+(L−) is not Legendrian isotopic to Sn

+(Lnt− ) and Sn

−(L+) is not Legendrian isotopicto Sn

−(Lnt+ ) for any n. Also, S+(Lnt

− ) is not Legendrian isotopic to S−(Lnt+ ).

Item 4 in this theorem has been established in the previous sections. We thus proveitems 1-3 through three lemmas.

Lemma 5.2.2 There are precisely four maximal Thurston-Bennequin representatives ofK ′, which we call L± and Lnt

± , and which have tb(L±) = tb(Lnt± ) = 2 and r(L±) = r(Lnt

± ) =±1. The L± are Legendrian divides on solid tori that thicken; the Lnt

± are Legendrian divideson solid tori that fail to thicken.

Proof. We know that any L at tb = 2 can be represented as a Legendrian divide on theboundary of a solid torus N representing K. Suppose N thickens; then it must thickento a standard neighborhood N ′ of a Legendrian (2, 3) torus knot with two dividing curvesand slope 1. This means that there are two possible tight contact structures on N , both


Figure 5.1: Shown is the Legendrian mountain range for the (1, 2)-cabling of the (2, 3) torusknot.

universally tight, and the extension to N ′ is a basic slice that is determined by the tightcontact structure on N (since there cannot be mixing of sign inside N ′). Now there are twochoices for tight contact structures on N , distinguished by the characteristic foliation ona convex meridian disc. However, once N ′ is determined (by the contact structure on N),then the tight contact structure on S3\N ′ is unique up to isotopy, since N ′ is the standardtubular neighborhood of the unique maximal tb representative of K. This proves that thereare at most two maximal tb representatives of K ′ coming from solid tori that thicken. Tosee that there are two, we calculate their rotation numbers, and show they are distinct.To see this, note that the two N that thicken have convex meridian curves with rotationnumbers r(∂D) = ∓1, and that the corresponding convex preferred longitudes will haverotation numbers r(∂Σ) = ±1. Thus r(L±) = 1(∓1) + 2(±1) = ±1.

Now suppose L at tb = 2 is a Legendrian divide on one of the N11 . Then the tight

contact structures on the complements of the two N11 are always contact isotopic, so there

are at most two Legendrian representatives; they are in fact distinct, since we know thatr(∂D) = ±1 and r(∂Σ) = 0 for N1

1 , which yields r(Lnt± ) = ±1. �

Lemma 5.2.3 If L represents K ′ and tb(L) < tb(K ′), then L destabilizes.

Proof. We may assume that L is a Legendrian ruling on a solid torus N representingK, and that L has efficient intersection with Γ∂N (for otherwise L will admit an obviousdestabilization). We have two cases. If slope(Γ∂N ) < 1, then inside N is a solid torus withboundary slope 1, and L will destabilize via a convex annulus joining the two boundarytori. On the other hand, if slope(Γ∂N ) > 1, then either inside N , or containing N , is a solidtorus with boundary slope ∞, and thus a convex annulus joining the two boundary tori willeither contain bypasses on the ∂N -edge, or will be standard convex. If the former, then Ldestabilizes; if the latter, then we may assume L is a Legendrian ruling on the solid toruswith boundary slope ∞, which thickens to a boundary torus of slope 1 since the associatedLegendrian representative of K destabilizes. Thus L will destabilize via a convex annulus.�

5.3. LEGENDRIAN SIMPLE ITERATED TORUS KNOTS 75

Lemma 5.2.4 S−(L−) = S−(Lnt− ) and S+(L+) = S+(Lnt

+ ); also S−(L+) = S+(L−).

Proof. Since L− is a Legendrian divide on N with boundary slope 2, S−(L−) is a Legendrianruling curve on ∂N ′ ⊂ N , where N ′ is a solid torus representing K with boundary slope∞. Similarly, since Lnt

− is a Legendrian divide on N11 with boundary slope 2, S−(Lnt

− ) isa Legendrian ruling curve on ∂N ′′ ⊂ N , where N ′′ is a solid torus representing K withboundary slope ∞. Now N ′ and N ′′ are neighborhoods of Legendrian knots representingK with tb = 0. If the associated rotation numbers are the same, then they are contactisotopic (by the Legendrian simplicity of the (2, 3)-torus knot). One may easily check thatthe rotation numbers are indeed the same, and thus there is an ambient contact isotopytaking N ′ to N ′′, and it simply remains to isotop S−(L−) to S−(Lnt

− ) through ruling curves.

A similar argument works for the positive stabilization case.

Finally, S−(L+) and S+(L−) are both Legendrian rulings on a standard neighborhoodof a unique tb(K) knot, and hence are Legendrian isotopic. �

5.3 Legendrian simple iterated torus knots

In this section we establish three well-defined families of Legendrian simple iterated torusknots; these follow immediately from the results either reviewed, or established, in Chapters3 and 4.

5.3.1 Iterated cablings of negative torus knots are simple

Negative torus knots are Legendrian simple [18]; this, combined with Theorem 3.2.1 andCorollary 4.1.2, yields the following theorem:

Corollary 5.3.1 All iterated cabling knot types that begin with negative torus knots areLegendrian simple; that is, if Kr = ((P1, q1), ..., (Pr, qr)) is an iterated torus knot type where(P1, q1) is a negative torus knot, then Kr is Legendrian simple.

5.3.2 Iterated cablings greater than w(Ki) are simple

The following is an immediate corollary of Theorem 3.2.7 and Section 4.3. However, thereis also a proof in [30] that was established independent of the work of Tosun. In particular,this latter proof establishes the specific shape of the Legendrian mountain ranges (whichactually follows from the proof of Tosun, as well).

Corollary 5.3.2 Let Kr = ((P1, q1), ..., (Pi, qi), ..., (Pr, qr)) be an iterated torus knot where(P1, q1) is a positive torus knot, and such that Pi+1/qi+1 > w(Ki) = tb(Ki) for 1 ≤ i < r.Then Kr is Legendrian simple and has a Legendrian mountain range with a single peak attb(Kr) = −χ(Kr) and r = 0.


5.3.3 Cablings less than lw(Ki), after repeated cablings greater than w(Ki),are simple

The following is an immediate corollary of Section 4.3, Theorem 4.1.3, and Corollary 4.1.2:

Corollary 5.3.3 Let Kr+1 = ((P1, q1), ..., (Pi, qi), ..., (Pr, qr), (Pr+1, qr+1)) be an iteratedtorus knot such that Pr+1 < 0, (P1, q1) is a positive torus knot and Pi+1/qi+1 > w(Ki) for1 ≤ i < r. Then Kr+1 is Legendrian simple and satisfies the UTP.

Chapter 6

Braid foliations and interlockingsolid tori

In this chapter we review background concerning braid foliation techniques and interlockingsolid tori, so as to be able to prove in the next chapter that all non-thickenables in the classof iterated torus knots have interlocking representations. Strictly speaking, we will includemore background material on braid foliations than is necessary; however, the additionalmaterial should give the reader a more complete picture of braid foliation techniques as awhole.

6.1 Braid foliations on embedded surfaces

In this subsection we recall tools first used by Bennequin [2], and further developed by Bir-man and Menasco [5, 3, 35], which allows the researcher to study braids via their relationshipto compact surfaces in S3.

6.1.1 The braid fibration of S3

The braid fibration of S3 is the standard open book decomposition of S3 consisting of anunknotted binding and discs as pages. In the braid setting, the unknotted binding willbe called the braid axis, and denoted by A; the disc pages will be denoted as a set byH = {Hθ|0 ≤ θ < 2π}.

A closed braid X, or simply a braid, in S3 is an oriented knot or link which intersectseach disc fiber Hθ transversely and positively, where here the orientation of each disc fiberHθ agrees with the orientation of A via a right-hand rule. For fixed θ-value, #(X ∩ Hθ) isthe braid index of X.

The braid fibration of S3 will be denoted by (S3, Hθ). We will be interested in studyingsurfaces embedded in (S3, Hθ), and these surfaces will usually have a braid X embedded inthem. We therefore need to recall terminology and facts used when studying surfaces in the

77

78 CHAPTER 6. BRAID FOLIATIONS AND INTERLOCKING SOLID TORI

braid fibration. Our main reference will be [3]; however, we will also draw from backgroundfound in [35].

6.1.2 Vertices and singularities

Let S be a compact surface embedded in the braid fibration, possibly with boundary com-ponents that are braids. We are interested in studying the braid foliation on S which isinduced by the braid fibration (S3, Hθ). To do this we first need some terminology.

A vertex in the braid foliation on S is a point v where A intersects S. A singularity inthe braid foliation on S is a point s where Hθ ⊂ TsS for some θ-value.

After identifying all vertices and singularities on S, the rest of the braid foliation on S isthen obtained by integrating the intersections of TxS with the Hθ at the θ-value for x ∈ S.For a fixed θ-value, any component of S ∩ Hθ is called a leaf in the foliation of S. Singularleaves contain a singularity; non-singular leaves do not.

6.1.3 Normalization of surfaces in the braid fibration

Let S be a compact surface, either with braided boundary components X, or closed withan embedded braid X. By definition, any braid X will be transverse to the braid foliationon S. Moreover, the following are true after isotopy of S relative to X:

• S intersects A transversely at finitely many points; i.e., there are finitely many ver-tices.

• The foliation at every vertex is radial; i.e., it looks like an elliptic point.

• All singularities are simple saddle points; i.e., they look like hyperbolic points. (Thepoint here is that local minima and maxima can be removed with isotopy of S relativeto the braid.)

• There are finitely many singularities.

• Any arc in the foliation connecting two different singularities must pass through avertex.

• Each non-singular leaf is an arc or simple closed curve.

• Each singular fiber contains exactly one singular leaf of the foliation; i.e., the singu-larities occur at distinct θ-values.

6.1.4 Three types of non-singular leaves

There are three types of non-singular leaves:

• a-arcs have one endpoint on a boundary component of S and one endpoint at a vertex.

6.1. BRAID FOLIATIONS ON EMBEDDED SURFACES 79

• b-arcs have two endpoints at two different vertices.

• c-circles are embedded simple closed curves.

Note that there are no non-singular arcs which have both of their endpoints on theboundary of S; such a scenario would cause the boundary to fail to be braided.

There is another notion that should be mentioned here. For a fixed θ-value, if a b-arccobounds (with a subarc of A) a subdisc of Hθ such that the braid X does not intersectthat disc, the b-arc is said to be an inessential b-arc. Similarly, if a c-circle bounds a discin Hθ which is not intersected by the braid X, that c-circle is said to be an inessentialc-circle. On the other hand, b-arcs or c-circles which are not inessential are simply said tobe essential.

6.1.5 Five types of singular leaves

Note that, moving forward through the fibration in the direction of increasing θ-coordinate,any singular leaf in the foliation is formed by non-singular leaves moving together to touch ata saddle singularity. After the saddle, the singular leaf is transformed into new non-singularleaves (see Figure 6.1). We will refer to this process as a surgery of the non-singular leaves,and label the singularities, singular leaves, and corresponding surgeries according to thenon-singular leaves associated to them to obtain five types of singular leaves:

• type aa: an a-arc surgered with an a-arc.

• type ab: an a-arc surgered with a b-arc.

• type bb: a b-arc surgered with a b-arc.

• type bc: a b-arc surgered with itself or a c-circle.

• type cc: a c-circle surgered with itself or another c-circle.

Figure 6.1 shows the five types of surgeries resulting in the five types of singularities;the singularities themselves occur in the middle frame for each three-frame Hθ sequence.

The following theorem is then a completed normalization theorem for a surface S in thebraid fibration:

Theorem 6.1.1 (Birman-Finkelstein) Let F be a surface with braided boundary; let Cbe a closed surface.

1. F can be chosen so that every non-singular leaf in its foliation is an a-arc or anessential b-arc, and every singularity in its foliation has type aa, ab, or bb.

2. C can be chosen so that every non-singular leaf in its foliation is an essential b-arcor an essential c-circle, and every singularity in its foliation has type bb, bc, or cc.


Figure 6.1: Shown are the five types of singular surgeries. For each type, the middle frame in theHθ sequence (dentoed by Hθ2

is the actual singular leaf. This figure is by Birman and Finkelstein.


6.1.6 Five types of tiles

As a consequence of the above theorem, any surface S can be decomposed into piecesby cutting along non-singular leaves so that each piece contains exactly one singular leaf,and has boundary components made up of non-singular leaves and possibly subarcs of theboundary of S. Such pieces are called tiles, and since there are five types of singular leaves,there are five types of tiles, pictured in Figure 6.2.

Figure 6.2: Shown are the five types of tiles that compose a surface S in the braid fibration; thearrows indicate the direction of increasing θ-coordinate around each vertex. The braid is indicatedby the heavy black arcs in the ab- and aa-tiles. This figure is by Birman and Finkelstein.

6.1.7 Parities of vertices and singularities

For an oriented surface S (where the orientation agrees with that of any of its bound-ary components), vertices and singularities in the braid foliation can be assigned a parity.Specifically, positive vertices v are such that the orientation of the braid axis A agrees withthe orientation TvS at v; negative vertices are such that those two orientations disagree.Similarly, positive singularities s are such that the orientation of Hθ and TsS agree at s,while negative singularities are such that those two orientations disagree.

We note here that in ab and aa tiles, the vertices which are joined to ∂S by a-arcs arepositive vertices; this can be seen by considering that X is braided positively with respectto the braid fibration.

An interior vertex is a vertex v that is not an endpoint for any a-arc. For such a vertex v,let star(v) be all singularities having v as an endpoint for their singular leaf. The followinglemma is an important one, as we shall see that it is very much related to the fact that(S3, ξstd) is a tight contact structure.


Lemma 6.1.2 (Birman-Finkelstein) Let v be an interior vertex. Then star(v) containsboth positive and negative singularities.

Note that the valence of a vertex v is the number of singular leaves that contain v.

6.1.8 The graphs G++, G−−, G−+, and G+−

For the moment assume that the braid foliation for S contains no c-circles; we will call sucha surface a tiling. Then by the normalization theorem above, the braid foliation for S willcontain four graphs that describe the skeleton of the braid foliation. If we let δ = ±, thenwe can define the graphs as follows:

• The graph G+,δ has as its edges those subarcs of singular leaves which join the two+ vertices in an aa, ab, or bb tile of parity δ. The vertices of G+,δ are the endpointsof its edges, along with all + vertices in the tiling of S which are not adjacent to anysingular point of sign δ.

• The graph G−,δ has as its edges those subarcs of singular leaves which:

– join the two negative vertices in a bb tile of sign δ.

– join the negative vertex in an ab tile of sign δ to ∂S.

– join ∂S to ∂S in an aa tile of sign δ.

The vertices of G−,δ are the endpoints of these edges, together with negative verticesin the tiling of S which are not adjacent to any singular point of sign δ.

Note that if one thinks of ∂S as acting like a negative vertex, then the definition of G−,δ

can be seen to be analogous to that of G+,δ.Subarcs of the graphs Gǫ,δ are shown in Figure 6.3 for the three types of tiles in this

subsection.

6.1.9 The graphs Gǫ,δ for closed surfaces

If S is a closed surface, then by Lemma 6.1.2, and the definition of the graphs Gǫ,δ, thefollowing facts hold:

Lemma 6.1.3 (Birman-Finkelstein)

1. Gǫ,δ ∩ G−ǫ,−δ = ∅.

2. Every singular point in the foliation of S is in G+,+ or G−,− (and so also in G+,− orG−,+).

3. Every vertex in the foliation of S is in G+,+ or G−,− (and so also in G+,− or G−,+).

4. Every vertex is the endpoint of an edge of some Gǫ,δ.

5. Gǫ,δ contains no closed loop bounding a disc on S.


Figure 6.3: Shown are subarcs of Gǫ,δ for aa, ab, and bb tiles. This figure is by Birman andFinkelstein.


6.1.10 The braid foliation on a tiling is the characteristic foliation Sξ

The braid fibration (S3, Hθ) supports the standard contact structure ξstd, which means thatξstd can be isotoped to be transverse to A and nearly tangent (on compact subsets) to thepages Hθ. It is then immediately evident that after such an isotopy, if a tiling is normalizedas above with respect to the braid fibration, then the characteristic foliations Sξ will beǫ-close to the braid foliation on S. We will thus usually make no distinction between thetwo. However, note that a characteristic foliation is not necessarily a braid foliation; this isa big enough distinction that we state it as a key principle:

Key Principle: The braid foliation for a tiling is its characteristic foliation; however, acharacteristic foliation is not necessarily a braid foliation.

6.1.11 Closed normalized surfaces with tilings are convex

Since the tiling on a normalized closed surface S is the characteristic foliation, and thattiling is Morse-Smale (see the normalization items above), we have that a normalized closedsurface S with a tiling in the braid fibration (S3, Hθ) is a convex surface. Furthermore, weknow that G++ must be contained in S+, while G−− must be contained in S−; furthermore,the dividing set ΓS will separate G++ from G−−.

We can also see that for a closed surface S, a closed loop in either G++ or G−− wouldimply a null-homotopic dividing curve, and hence an overtwisted disc.

6.1.12 Braid foliations on tori

Let T be a torus with an embedded braid X. Since T is closed, after normalization allnon-singular leaves in the braid foliation on T will consist of either b-arcs or c-circles.

If the braid foliation for T is composed solely of non-singular leaves, all of which arec-circles, then T is said to have a circular foliation.

If the braid foliation for T contains both non-singular b-arcs and c-circles (and hencemust have singularities), then T is said to have a mixed foliation.

If all of the non-singular leaves in the braid foliation for T are b-arcs, then the braidfoliation for T is said to be a tiling. Notice that for a tiling, G++ and G−− will be graphsthat deformation retract onto simple closed curves; the dividing set will be parallel push-offsof these simple closed curves.

A tiling which consists solely of valence-4 vertices is said to be a standard tiling.

6.1.13 Braid moves on foliated surfaces

One of the main advantages of studying braids X embedded in surfaces S is that thebraid foliation on S can be used to identify admissible braid moves for X. In particular,stabilization/destabilization and exchange moves have a natural interpretation in terms ofthe braid foliation.


Destabilization

Figure 6.4 shows a destabilizing disc for a braid X on a surface S. If v is of positive parity,then X is oriented from right to left, and the parity of s determines the parity of thedestabilization: a negative sign for s results in a negative destabilization, a positive signresults in a positive destabilization.

Figure 6.4: Shown is a destabilizing disc for the braid X; the braid currently is in heavy black, andafter destabilization it is dashed. The parities of v and s determine the parity of the destabilization.

Stabilization can obviously be performed over the same disc, but with the starting andending positions of the braid X reversed.

Exchange move

Figure 6.5 shows an exchange move for the braid X on the surface S; the key here is thats0 and v0 are positive, while s1 and v1 are negative.

6.1.14 Manipulation of the braid foliation

Suppose X is embedded in a closed surface S. There are two standard ways of manipulatingthe braid foliation, and one non-standard way, which we will review in this subsection.

Standard change of foliation

The standard change of foliation is sometimes called a standard change of fibration; theidea is that the change can either be thought of as the result of an isotopy of the surface,or an isotopy of the braid fibration. The following lemma specifying the standard changeof foliation appears in [35]; the key is that there are two adjacent singularities of the samesign:


Figure 6.5: Shown is an exchange move of the braid X on the surface S; the heavy black line is thebraid before the exchange move, the dashed line the braid after. The vertex and singularity v0 ands0 are positive; v1 and s1 are negative.

Lemma 6.1.4 (Menasco) Let R be the closure of a region foliated by b-arcs such thatthere are vertices v+, v− ⊂ ∂R and singularities s1, s2 ⊂ ∂R where s1, s2 have the sameparity. Then there exists a change of foliation such that the valence of these two verticeshas been decreased.

Figure 6.6 shows the picture of how this change of foliation appears on the surface S;there are two cases, depending on particular embeddings of the knot K (our braid X) withrelation to the region in question.

Figure 6.6: Shown is the standard change of foliation, which can always be performed wheneverthere are two adjacent singularities of the same sign. This figure is by Menasco.


Figure 6.7 shows the actual local isotopy of S that results in the standard change offoliation; this isotopy can be thought of as simply moving one singularity past the other inthe positive θ-direction. Notice that this transposition of the order of singularities can bethought of as an isotopy of the surface S through a topological disc.

Figure 6.7: Shown is a local picture of the embedding of the region R that is involved in thestandard change of foliation; the actual isotopy of the surface that results in the standard changeof foliation consists of moving one of the singularities past the other in the θ-coordinate. In thispicture the knot does not appear, for simplicity. This figure is by Birman and Finkelstein.

Elimination of valence two vertex

In this lemma, star(v0) is the closure of all non-singular b-arcs having an endpoint at thevertex v0.

Lemma 6.1.5 (Menasco) Let v0 be a valence two vertex, with star(v0) topologically adisc. Suppose star(v0) contains vertices v1, v2 and singularities s+, s− of opposite sign.Then, after an isotopy of X involving only exchange moves, destabilizations, and braidisotopies, we can locally eliminate vertices v0 and v1 and singularities s+ and s− in thefoliation of S.

Figure 6.8 shows the before and after picture on the surface S, again with two cases de-pending on the particular embedding of the braid X with relation to the region in question.

Figure 6.8: Shown is the elimination of a valence two vertex. This figure is by Menasco.


Figure 6.9 again shows the actual isotopy of the surface S that results in the eliminationof the two vertices; one can see that it can be thought of as the removal of a pocket, and toaccomplish this removal one may have to perform exchange moves.

Figure 6.9: Shown is the isotopy of the surface S that results in the elimination of a valence twovertex. Exchange moves may have to be performed to allow for this removal. This figure is byBirman and Finkelstein.

Non-standard change of foliation

As opposed to the previous two standard manipulations of the braid foliation, which canalways occur given the right conditions, the following non-standard change of foliationcannot always be accomplished: either the embedding of the surface itself may prevent it,or the embedding of the braid X on the surface S can prevent it.

The non-standard change of foliation involves two adjacent singularities of oppositesigns; in short, an isotopy of the surface rearranges their order in the θ-coordinate. Thereare actually four ways this can happen, two that occur when the negative singularity occursbefore the positive singularity in the θ-ordering, and two that occur when the positivesingularity occurs before the negative singularity in the θ-ordering. Figure 6.10 shows thetwo types that can occur when the negative singularity occurs before the positive singularityin the θ-ordering; the picture shows the local change in the braid foliation on S.

Figure 6.11 shows the isotopy of the surface S that occurs in order to bring about theType II change of foliation in Figure 6.10. Note that the presence of a braid X on the surfacecan prevent this change of foliation from occurring, as the change of foliation violates thetransversality of the braid. We also note that the embedding of the surface S can preventa non-standard change of foliation, as we will see in the next section when we examineinterlocking solid tori.


Figure 6.10: Shown are the two types of non-standard changes of foliation that can occur when anegative singularity occurs before an adjacent positive singularity in the θ-ordering.

Figure 6.11: Shown is an embedding and isotopy of a local region of S that results in a non-standardchange of foliation. Note that the presence of a braid on the surface S may prevent the non-standardchange of foliation from occurring. The actual embedding of the surface itself may also prevent anon-standard change of foliation from occurring, as we will see in the next section.


6.1.15 Non-standard changes of foliation result from bypasses

We can now present another significant connection between braid foliations and convexsurface theory. In particular, we have the following fact:

• Any non-standard change of foliation is accomplished by isotopy of the surface Sthrough a bypass off of G+− or G−+.

This can be seen by carefully examining a picture of the embedding of the region affectedby a non-standard change of foliation; see Figure 6.12.

Figure 6.12: Shown is the embedding for a non-standard change of fibration and the indicatedbypass attachment.

6.2 Interlocking solid tori: An obstruction to simplicity

In this section we review necessary conditions for transversal non-simplicity in the class ofiterated torus knots, as discovered by Menasco in [37]. This will lead us to a definition ofinterlocking solid tori, the objects of interest in the next chapter.

6.2.1 Exchange reducible knots

Let X be a braid representation of a knot type K; then X has a braid index, say nX . If welet X vary over all braid representations of K, then there will be a minimum braid indexfor the knot type K, denoted nmin.

A knot type K is said to be exchange reducible if any braid representation X is iso-topic to some braid representation Xmin at minimum braid index nmin by a sequence ofdestabilizations, exchange moves, and braid isotopies. This sequence of braid moves is thusnon-increasing on braid index.

6.2.2 Exchange reducibility implies transversal simplicity

The following result of Birman and Wrinkle shows why exchange reducible braids are in-teresting from our vantage point [8]:

6.2. INTERLOCKING SOLID TORI: AN OBSTRUCTION TO SIMPLICITY 91

Theorem 6.2.1 (Birman-Wrinkle) If K is exchange reducible, then K is transversallysimple.

The converse is not true; in fact counterexamples can be obtained, with some argument,from our results for iterated torus knots.

6.2.3 Exchange reducibility in the class of iterated torus knots

Menasco has studied exchange reducibility in the class of iterated torus knots [35, 36, 37].In particular, the following theorem holds:

Theorem 6.2.2 (Menasco) Torus knots are exchange reducible, and hence transversallysimple.

To prove this theorem, Menasco studied torus knots represented as braids on embeddedsolid unknotted tori in the braid fibration (S3, Hθ). Menasco then extended these techniquesto the class of iterated torus knots to establish the following, where X is a braid representingan arbitrary iterated torus knot Kr, and Tr−1 is the cabling torus:

• If it is the case, for arbitrary X representing arbitrary Kr, that after a sequence ofdestabilizatons, exchange moves, and braid isotopies of X, Tr−1 can be assumed tohave a circular foliation, then iterated torus knots are exchange reducible.

• It is true that if the above Tr−1 has a mixed foliation, then it can be assumed, after asequence of destabilizatons, exchange moves, and braid isotopies of X, to be a circularfoliation.

• It is not true that if Tr−1 is a tiling, that it can be made to be a mixed foliation. Inparticular, if Kr fails to be exchange reducible, then the following must be true:

– For i < r, there must exist a standard tiling Ti representing Ki.

– Ti must have a uniform steps configuration that has homogeneous twisting.

– Ti must have a uniform steps configuration that is interlocking.

– Ki+1 must satisfy particular conditions with respect to the graphs G++, G−−, G+−,and G−+.

Our goal in the rest of this section is to explain these last itemized points, in partic-ular to elucidate the definitions of uniform steps configuration, homogeneous twisting, andinterlocking. Before doing so, we briefly explain the argument behind the above items. Totry to reduce a tiling Tr−1 to a mixed foliation, Menasco put a complexity measure on thetiling based in part on the number of vertices. After standard changes of foliation thatreduced valence three vertices to valence two vertices, and after elimination of valence twovertices, Tr−1 could be assumed to be a standard tiling, with each vertex having valencefour, and no adjacent singularities of the same sign. Menasco then looked for the existenceof non-standard changes of foliation, and found that they could be obtained unless all ofthe above conditions held; in the subsections that follow, we will see why this is the case.


6.2.4 Homogeneous twisting uniform steps configurations

The solid tori which we will construct will have a number of properties; we begin with themost basic, and work toward the more complex:

Block-disc presentations of a braid X

The solid tori we construct will represent a braid X – so here the braid X is the core ofthe solid torus N . This braid will project onto a cylinder of radius r0, and thus we willuse cylindrical coordinates to describe the construction, where z is a cyclic coordinate withperiod 1 and θ is a cyclic coordinate with period 2π. A block-disc presentation of a braidX will consist of the following:

• m circular discs, all of the same radius r0, stacked at m different z-levels so that thebraid axis A intersects each of the discs once. The discs will be labeled D1, · · · , Dm.Their respective z-coordinates will be given as z1, · · · , zm.

• m rectangles (topological discs), denoted R1, · · · , Rm. We will refer to these rectanglesas blocks. We will denote the top, bottom, right, and left edges of an Ri as Rt

i, Rbi ,

Rri , and Rl

i, respectively; these edges will have fixed z or θ-coordinates indicated byzti > zb

i , and θri > θl

i, respectively. We then embed each Ri as follows:

– Each Ri has a braid foliation that is simply arcs parallel to Rri , and Rl

i; also,the z-level sets of Ri are arcs parallel to Rt

i and Rbi ; in both cases, each arc

corresponds to a unique value of θ or z, respectively.

– Rti and Rb

i are each glued to the boundary of a separate disc.

– The r-coordinate of any point on Ri\(Rti ∪ Rb

i ) is greater than r0 (so blocks goover discs).

• Every disc Di is glued to exactly two blocks Rj and Rk, and the block-disc complexdeformation retracts onto a braid X.

We note here that a regular neighborhood of a block-disc presentation of a braid X isa set of solid tori (one for each component of the link X) whose boundary tori have braidfoliations that are tilings. To clarify, associated to each disc Di will be two vertices, v+

i andv−i (at the top and bottom of the boundary of a neighborhood of the disc), and associatedto each block will be two singularities, s+

i and s−i (at the right and left edges of the regularneighborhood of the block). Furthermore, there are no c-circles in the braid foliation. SeeFigure 6.13.

Uniform steps configuration

Once we have a block-disc presentation of a braid X, we can order the blocks Ri cyclicallywith respect to their position as we positively traverse the braid. With this ordering in


Figure 6.13: Shown are the two vertices, one positive and one negative, associated with a disc, andthe two singularities, one positive and one negative, associated with a block.

mind, a uniform steps configuration is a block-disc presentation of a braid X that has thefollowing property:

• There exists a positive integer k < m so that θri = θl

i+k for all i; i.e., the right edgeof the i-th block has the same θ-coordinate as the left edge of the (i + k)-th block,computed modulo m.

We call the integer k the step size for the steps configuration. See Figure 6.14.

Figure 6.14: Shown is a uniform steps configuration for step size k = 2.

We note here that a regular neighborhood of a uniform steps configuration of a braidedknot X is a solid torus whose boundary torus has a braid foliation that is a standard tiling.

Homogeneous twisting

Further conditions can be placed on a uniform steps configuration that restrict the knottypes which can be represented. In particular, a uniform steps configuration is said to be


positive twisting if it has the following property, where the ordering of the blocks is againcoming from the braid-induced ordering:

• zti = zb

i+1 for all i; i.e., the top edge of the i-th block is at the same z-level as thebottom edge of the i + 1-st block.

Similarly, a uniform steps configuration is said to be negative twisting if:

• zbi = zt

i+1 for all i; i.e., the bottom edge of the i-th block is at the same z-level as thetop edge of the i + 1-st block.

If the uniform steps configuration is either positive twisting or negative twisting, thenit is said to have homogeneous twisting.

Figure 6.15 shows a local picture of both negative twisting (a) and positive twisting (b).

Figure 6.15: Shown in part (a) is negative twisting, with step size k = 3; in part (b) is positivetwisting with step size k = 2. The gray rectangles are blocks; the horizontal lines represent theboundaries of discs for an interval of θ-values.

6.2.5 The G-framing

Figure 6.16 shows local pictures of negatively twisting and positively twisting steps configu-rations. In the negative case, G++ is shown; in the positive case, G+− is shown. Continuingthis local picture over the whole negative or positive steps configuration for a braid X letsus observe the following, provided X is a knot:

• G++ is a longitude for the boundary torus of any negatively twisting uniform stepsconfiguration.

• G+− is a longitude for the boundary torus of any positively twisting uniform stepsconfiguration.

We will call the framing with this longitude the G-framing for the uniform steps config-uration; context (whether we are looking at negative twisting or positive twisting) will tellus whether we actually mean G++ or G+−.


Figure 6.16: Shown in (a-) and (b-) is a local picture for a negatively twisting steps configuration;similarly, (a+) and (b+) show a local picture for a positively twisting steps configuration. The (a)’sindicate G++ or G+−; the (b)’s indicate the longitude in the G-framing for that torus.

6.2.6 Calculating boundary slopes

Consider a negatively twisting uniform steps configuration; the previous subsections showsthat G++ is a longitude for the resulting torus. Moreover, one can observe that this torus,when thought of as convex, has exactly two dividing curves parallel to G++. This yieldsthe following fact:

• Negatively twisting uniform steps configurations are standard neighborhoods of Leg-endrian knots.

On the other hand, a positive twisting uniform steps configuration may have multipledividing curves. However, the intersection boundary slope is easy to compute in the G-framing. Figure 6.17 shows a local picture from which one can derive the following fact:

• Let ∂N be a boundary torus for a positively twisting uniform steps configuration withstep size k and m blocks. Then slope(Γ∂N ) = −(k + 1)/m in the G-framing.

6.2.7 Interlocking, homogeneous twisting, uniform steps configurations

Consider a positive twisting uniform steps configuration with step size k. Associated witheach block Ri is the z-value zb

i (the least z-value for that block); likewise, associated to theblock Ri+k is the z-value zt

i+k (the greatest z-value for that block). We then associate with

the value θri = θl

i+k an arc λi such that λi has constant θ-value of θri = θl

i+k and z-values

that traverse the closed interval [zbi , z

ti+k]. We will relabel zb

i = z−,bλi

and zti = z−,t

λi, and we


Figure 6.17: Shown is a local picture of a positively twisting uniform steps configuration. In heavygray is a G-longitude. In heavy black are subarcs of G++ (or subarcs of dividing curves); it is evidentfrom the picture that the intersection of G++ with any meridian will be k +1, while the intersectionnumber with the G-longitude is equal to the number of blocks.

will likewise relabel zbi+k = z+,b

λiand zt

i+k = z+,tλi

; we will refer to the θ-coordinate of λi asθi. Note that the z-support of a λ-arc contains the z-support of k + 1 consecutive blocks.Figure 6.18 shows a picture of a λ-arc with the four z-levels indicated.

The positive twisting uniform steps configuration with step size k is said to be (positively)interlocking if the following property holds:

• For consecutive λi and λi+1 in the braid ordering as we move forward along the braidX, there exists a sequence λi1 , · · · , λin such that:

– λi1 = λi and λin = λi+1.

– θi = θi1 < θi2 < · · · < θin = θi+1.

– After cyclic translation, z−,tλij

> z+,tλij+1

> z−,bλij

> z+,bλij+1

.

Figure 6.19 shows this positive interlocking sequence; the intuitive idea is that if we tryto push the right edge of the i-th block past the right edge of the (i+1)-st block, we cannot,since the embedding of the solid torus gets in the way of itself. The λ-arcs will be calledinterlocking λ-arcs.

To see how a negatively twisting uniform steps configuration can be interlocking, we dosomething similar. Associated with each block Ri is the z-value zt

i (the greatest z-valuefor that block); likewise, associated to the block Ri+k is the z-value zb

i+k (the least z-value

for that block). We then associate with the value θri = θl

i+k an arc λi such that λi has


Figure 6.18: Shown is a λ-arc with the various z-levels labelled.

Figure 6.19: Shown is a positive interlocking sequence; the blocks are shown, the discs are implied.Intuitively, as we try to push the λi arc forward, it runs into successive λ-arcs, eventually runninginto λi+1. Notice that the gray block at the top with the two tick marks is identified with the grayat the bottom with two tick marks, and therefore λi3 is continued from the bottom to the top aswell.


constant θ-value of θri = θl

i+k and z-values that traverse the closed interval [zbi+k, z

ti ]. We

will relabel zti = z+,t

λiand zb

i = z+,bλi

, and also relabel zti+k = z−,t

λiand zb

i+k = z−,bλi

and referto the θ-coordinate of λi as θi.

The negative twisting uniform steps configuration with step size k is said to be (nega-tively) interlocking if the following property holds:

• For consecutive λi and λi+1 in the braid ordering as we move forward along the braidX, there exists a sequence λi1 , · · · , λin such that:

– λi1 = λi and λin = λi+1.

– θi = θi1 < θi2 < · · · < θin = θi+1.

– After cyclic translation, z+,bλij

< z−,bλij+1

< z+,tλij

< z−,tλij+1

.

Figure 6.20 shows this negative interlocking sequence; the intuitive idea again is that ifwe try to push the right edge of the i-th block past the right edge of the (i + 1)-st block,we cannot, since the embedding of the solid torus gets in the way of itself.

Figure 6.20: Shown is a negative interlocking sequence; the blocks are shown, the discs are implied.Intuitively, as we try to push the λi arc forward, it runs into successive λ-arcs, eventually runninginto λi+1. Notice that the gray block at the top with the two tick marks is identified with the grayat the bottom with two tick marks, and therefore λi3 is continued from the bottom to the top aswell.

The connection between interlocking solid tori and the UTP will be made rigorous inthe following section; however, we can already begin to understand why they might berelated. In order for a solid torus to be non-thickenable, it must be embedded with respectto the ambient contact structure in such a way that prevents bypasses from being attached


to its exterior. From the braid foliation perspective, this means that the solid torus mustbe embedded so as to prevent bypasses off of G+− or G−+ in particular; i.e., there cannotbe non-standard changes of foliation that can be accomplished by isotoping the solid torusoutward. In order to prevent such non-standard changes of foliation, the embedding of thesolid torus itself must block such moves; interlocking blocks accomplish this obstruction.

6.2.8 An interlocking solid torus representing the (2, 3) torus knot

We show in Figure 6.21 the initial example of an interlocking, positively twisting, uniformsteps configuration representing the (2, 3) torus knot, discovered by Menasco and Matsuda[37].

Figure 6.21: Shown is a positively interlocking steps configuration for the (2, 3) torus knot, withstep size k = 1. The blocks on the far right and far left of the diagram wrap around the back; theblocks at the top and bottom of the diagram are identified.

In the next chapter we will establish the existence of such interlocking solid tori forall positive torus knots and for all step sizes k, and furthermore show that they preciselycoincide with non-thickenable solid tori. We will then complete a similar analysis for alliterated torus knots which fail the UTP.

6.2.9 Legendrian rectangular diagrams

In this final subsection we recall a way of representing Legendrian knots as (braided) rect-angular diagrams; our main reference is [34].

We begin with an example. If we examine G++ on the interlocking solid torus fromFigure 6.21, we obtain Figure 6.22, where the hyperbolic singularities are indicated by anx, and the elliptic singularities are indicated by a dot. Note that G++ is a Legendrian knot.

As noted in [34], there is a Legendrian isotopy of G++ that allows it to be braidedwith respect to the braid axis. Specifically, we can slightly perturb the Legendrian knotrepresenting G++ off of the braid axis so that it has monotonic winding with respect to


Figure 6.22: Shown is G++ on the k = 1 interlocking solid torus representing the (2, 3)torus knot.

the axis. The resulting knot then can be represented as a braided Legendrian rectangulardiagram as in Figure 6.23.

Note that this braided Legendrian rectangular diagram is composed of horizontal andvertical arcs; however, these arcs actually represent nearly horizontal and nearly verticalarcs, respectively. What this means is that the horizontal arcs are actually Legendrian arcswith a slightly negative slope on a cylinder of small radius r away from the braid axis, whilethe vertical arcs are actually Legendrian arcs with a highly negative slope on a cylinder oflarge radius r away from the braid axis. Consideration of the contact structure pictured inFigure 2.2 will show that such arcs are indeed Legendrian. Each point where a horizontaland vertical arc meet in the rectangular diagram thus represents a radial Legendrian arc atconstant z- and θ-levels.

In [34], it is shown that any such diagram represents a Legendrian knot, and everyLegendrian knot can be Legendrian isotoped to be in such a form. Furthermore, there areReidemeister moves that allow for Legendrian isotopy of Legendrian rectangular diagrams;these are picture in Figure 6.24. Of particular importance to us will be Legendrian horizontaland vertical flips, which will allow us to visualize certain Legendrian isotopies associatedwith Legendrian knots on interlocking solid tori. In particular, after horizontal flips, anyLegendrian rectangular diagram can be braided.


Figure 6.23: Shown is a braided Legendrian rectangular diagram representing G++ on thek = 1 interlocking solid torus representing the (2, 3) torus knot.

Figure 6.24: Shown are Reidemeister moves for Legendrian isotopy of Legendrian rectan-gular diagrams.


Chapter 7

Results: Interlockingnon-thickenables

In this chapter we investigate positively twisting interlocking steps configurations. Our maingoal is to show that every non-trivial non-thickenable solid torus in the class of iteratedtorus knots can be represented by an interlocking steps configuration; we also prove cablingtheorems that show that existence of interlocking steps configurations is preserved undercertain admissible cablings.

7.1 Interlocking non-thickenables for positive torus knots

In this section we prove the following theorem:

Theorem 7.1.1 Let Nk1 be a non-trivial non-thickenable representing a (p, q) torus knot.

Then Nk1 is represented by a positively twisting interlocking steps configuration with step

size k and (pqk + p + q) blocks.

The proof will proceed as follows: We first show how to construct such an interlockingsolid torus, and then show that its complement in S3 is precisely that of a non-thickenableNk

1 . Sections 7.1.1 to 7.1.5 will contain the proof.

7.1.1 The construction

The idea in this subsection is to generalize the construction of the interlocking positivelytwisting uniform steps configuration representing the (2, 3) torus knot in Figure 6.21.

Given a positive torus knot K = (p, q), we show how to construct a positively inter-locking steps configuration for arbitrary step size k representing K, and we will refer to theresulting solid torus as Nk

Int. Later on in this section we will then prove that NkInt = Nk

for all k; i.e., the NkInt are precisely the non-thickenable solid tori for positive torus knots.

We begin with (pqk + p + q) parallel stacked discs, each punctured once by the braidaxis. We then show how to attach blocks to these discs in order to achieve an interlocking

103

104 CHAPTER 7. RESULTS: INTERLOCKING NON-THICKENABLES

steps configuration. For convenience we will choose (pqk + p + q) θ-values, labelled θ0 <θ1 < · · · < θ(pqk+p+q−1), and thus think of our steps configuration as projecting ontoa (pqk + p + q) × (pqk + p + q) grid; the grid itself projects onto an unknotted torus,with left and right sides identified, and top and bottom sides identified. The discs will berepresented by the horizontal lines on this grid, with endpoints identified to indicate theircircular nature, and we will describe how to lay blocks onto the grid in order to achieve aninterlocking steps configuration. We will label the horizontal lines from top to bottom, withz0 > z1 > · · · > z(pqk+p+q−1). See Figure 7.1; this figure can be referred to for the next fewparagraphs.

With our grid in mind, each block Ri will have height p and width qk. It will be usefulat the outset to define a (k + 1)-stack of blocks to be a stack of (k + 1) positively twistingblocks Ri with uniform step size k. Note that a (k + 1)-stack contains one full λ-arc, andin fact can be characterized by that λ-arc.

We first construct the left edge of our grid. We begin at the top left corner of our grid,and place a (k + 1)-stack with a top block that has its left edge at θ0 and its top edge atz0. The k blocks below will therefore wrap around to the right side of the grid. The rightedge of the bottom block in the (k + 1)-stack will be at θ0; we will therefore have one λ-arcat θ0; we will call it λθ0 .

We next place another (k + 1)-stack so that the left edge of its top block is at θ1 andthe top edge of the same block is at z(pk+1) (this is one horizontal line below the top of thebottom block in the first (k + 1)-stack). Again, the k blocks below this top block will wraparound to the right side of the grid. We now have another λ-arc, namely λθ1 – we note thatby construction λθ0 and λθ1 are interlocking.

We continue this, placing the next (k + 1)-stack so that the top edge of its top blockis (pk + 1) units below the top edge of the previous (k + 1)-stack. We stop temporarilyafter placing down q number of (k + 1)-stacks. By construction, we will have λθ0 throughλθq−1 being interlocking, and the bottom edge of the last (k + 1)-stack will be at the levelz(pqk+p+q−1) (i.e., (q(pk + 1) + p) below the top horizontal line). Figure 7.1 shows thecompleted left edge construction for the (3, 4) torus knot and k = 2.

We have thus built the left edge of our grid by placing down entire (k + 1)-stacks (andby default the right edge as well). We now show how to continue the construction acrossthe grid by placing down single blocks to make new (k+1)-stacks. To see how to do this, weplace a single block with its top edge at a z-level that is (pk + 1) below the top edge of thelast (k + 1)-stack, and with its left edge at θq. This block will wrap around to the top andhave its bottom edge attach to the top of the original (k+1)-stack, and thus the left edge ofthe new block will represent a λθq

for a new (k+1)-stack. Note that by construction we stillhave λθq−1 and λθq

interlocking. We continue this procedure iteratively: if a (k + 1)-stackat λθi

has been completed, we then place a new block down with its left edge at θi+1 andits top edge (pk + 1) (cyclic) units down from the top edge of the last block placed. Byconstruction, λθi

and λθi+1 will be interlocking. Moreover, since (pk + 1) and (pqk + p + q)are relatively prime, this iterative procedure will result in every disc being identified withthe top of a unique λ-arc, and the process will terminate once this is accomplished. The

7.1. INTERLOCKING NON-THICKENABLES FOR POSITIVE TORUS KNOTS 105

Figure 7.1: Shown is the completed left side construction for the interlocking steps configurationrepresenting the (3, 4) torus knot with step size k = 2. The horizontal lines represent discs; thedashed vertical lines are θ-values. The dark black arcs represent interlocking λ-arcs.

final λ-arc will have its bottom point at the z-level zp−1, and will thus interlock with λθ0 ;as a consequence the construction is interlocking.

Moreover, the G-longitude for the steps configuration has a constant slope of p/q; i.e.,over q units and up p. Thus the braid resulting from the steps configuration will embedon an unknotted torus, and since p and q are relatively prime, the steps configuration willrepresent a (p, q) torus knot. Figure 7.2 shows the completed construction for the (3, 4)torus knot and k = 2.

7.1.2 Boundary slopes

It is easy to see that the G-framing for an NkInt corresponds to the C′ framing. Thus, since

an NkInt has (pqk+p+q) blocks, the boundary slope is slope(Γ∂Nk

Int) = −(k+1)/(pqk+p+q);

i.e., precisely the boundary slopes of the non-thickenables. However, this does not show thatthe Nk

Int fail to thicken; we must show that their complements in S3 are the complementsof non-thickenables. We do so in the next three subsections.

7.1.3 Standard neighborhoods of Legendrians

Given an interlocking steps configuration NkInt representing a positive torus knot (p, q), we

will show in this subsection how to construct a standard neighborhood of a Legendrianunknot in the complement of Nk

Int, which (in the next subsection) we will show to beN(L2) from the construction of non-thickenables Nk. The standard neighborhood that we


Figure 7.2: Shown is the completed interlocking steps configuration for the (3, 4) torus knot andk = 2.

construct will be a negatively twisting uniform steps configuration with step size (p − 1),and we will denote the steps configuration by N(L).

The construction is straightforward: Referring to the steps configuration for NkInt, we

now place (pqk + p + q) new discs that interleave with the original discs. We then attach(pqk + p + q) new blocks in the following manner. First, the top and bottom of each blockwill be attached to consecutive discs in the z-ordering. Thus each block Ri will have z-support that contains exactly one of the original discs; call this original disc Di. Then theθ-support of the block will be the exact complement of the θ-support of the blocks attachedto Di. Figure 7.3 shows a local picture.

Figure 7.3: Shown is a local slice of an NkInt with transparent blocks. In its complement is N(L),

with blocks in dark gray.

It is evident that N(L) deformations retracts onto a (−(qk + 1), 1) torus knot (as thereare qk blocks on the left edge of our diagram for Nk

Int, each of which supports one meridianof N(L), plus an extra p discs which will support the extra meridian). Since there are two


Legendrian divides with slope ∞′ in the C′ framing, a change of coordinates to the C framingyields slope(∂N(L)) = −1/(qk + 1). Thus N(L) has the same boundary slope as N(L2).

A complete picture of both NkInt and N(L) is shown in Figure 7.4 for the (3, 4) torus

knot and k = 2.

Figure 7.4: Shown is NkInt with N(L) for the (3, 4) torus knot and k = 2. Nk

Int has transparentblocks; N(L) has dark gray blocks.

7.1.4 Cabling annuli

We now show that there is a standard convex annulus in the complement of NkInt which

has its boundary components being ∞′ rulings. To see this, first note that for ∂NkInt, the

two simple closed curves G+− and G−+ are ∞′ rulings, as they are both Legendrian rulingsafter making ∂Nk

Int standard form convex, and we know that the G-framing corresponds tothe C′ framing.

Furthermore, due to the interlocking construction of NkInt, if we focus our attention on

one singular arc of G−+, it interlocks with a singular arc from G+−, as in the top portionof Figure 7.5.

Now we can adjust the θ- and z-levels so that in fact these two singular arcs from G−+

and G+− are I-invariant copies of each other. Thus there is a standard convex annulus Athat has Legendrian boundary on G+− and G−+, but has interior in the complement ofNk

Int. We show a portion of this annulus in the bottom portion of Figure 7.5.

After a horizontal Legendrian flip of the rulings on this standard convex annulus, wecan see that N(L) is in fact precisely one component of the complement of Nk

Int ∪ A. SeeFigure 7.6 which shows the example of the (3, 4) torus knot with k = 2.


Figure 7.5: The top portion shows interlocking singular arcs from G−+ and G+−; the bottomportion shows how there is a standard convex annulus joining G−+ and G+−.

Figure 7.6: Shown is an outline of NkInt with N(L) in gray and just a portion of the cabling annulus

in darker gray; this is for the (3, 4) torus knot with k = 2. The reader is encouraged to extrapolateto the whole cabling annulus and observe that N(L) is precisely one component of the complementof Nk

Int ∪ A.


7.1.5 Verification of non-thickening: the second standard neighborhood

We saw in the previous section that one component of S3\(NkInt ∪A) is N(L), the standard

neighborhood of a Legendrian unknot with tb = −(qk + 1). We now show that the othercomponent of S3\(Nk

Int ∪ A) is the standard neighborhood of a Legendrian unknot withtb = −(pk + 1); this will complete the proof that the Nk

Int are precisely the non-thickenableNk discovered using convex surface theory. (One will note that for a fixed (p, q) and k, oneinterlocking steps configuration will yield the two non-thickenable solid tori with boundaryslope −(k + 1)/(pqk + p + q) by a simple reversal of orientation of the solid tori; i.e.,diffeomorphism by −id.)

So it remains to identify the dividing set on ∂(S3\(NkInt ∪ A)). A local picture of the

intersection of ∂NkInt with A is shown in Figure 7.7. Parts (a) and (b) both show how

subarcs of the dividing curve on ∂NkInt that is the positive transverse push-off of G++

are connected to dividing curves on A that join hyperbolic singularities on ∂NkInt, all via

edge-rounding.

Figure 7.7: Parts (a) and (b) both show how subarcs of the dividing curve on ∂NkInt that is the

positive transverse push-off of G++ are connected to dividing curves on A that join hyperbolicsingularities on ∂Nk

Int, all via edge-rounding. The dividing curves are in heavy black.

We can get an idea of how this translates over the whole steps configuration if we lookat a larger slice of such an Nk

Int. This is shown in Figure 7.8.One can see that this edge-rounding will result in one of the two dividing curves on

∂(S3\(NkInt ∪ A)), as the edge-rounding for the annulus connects all arcs of the dividing


Figure 7.8: Shown is a slice of an NkInt with accompanying A and the edge-rounding of some

dividing curves.

curves that correspond to the λ-arcs in the interlocking steps configuration. Furthermore,one sees that the dividing curves have slope −1/(pk + 1), as the left-most bottom blockwill support (k + 1) longitudes, while the remaining p − 1 bottom blocks will support klongitudes each. See Figure 7.9 for the example of the (3, 4) torus knot with k = 2.

7.2 Cabling theorems for positive interlocking solid tori

In this section we prove a cabling theorem for positively twisting interlocking steps config-urations. We first introduce some notation. As usual, given a simple closed curve on thesurface of one of our cabled interlocking constructions, we will denote the algebraic inter-section with the ∞′ longitude as p. The new notation we will be using is that the algebraicintersection of the same curve with the G-longitude will be denoted pG. We then have thefollowing theorem:

Theorem 7.2.1 Let NkInt be a positively twisting interlocking steps configuration repre-

senting a knot K, where k is the step size, and mk is the number of blocks. Supposeqk′ = k for positive integers k′ and q, where q > 1. If pG is relatively prime to q, andpG/q > −mk/(k + 1), then there exists a positively twisting interlocking steps configurationNk′

Int,(pG,q)with step size k′ representing a (pG, q)-cabling of K in the G-framing. Moreover,

NkInt = Nk′

Int,(pG,q)∪N(L) ∪N(A), where N(L) is a standard Legendrian neighborhood of a

core curve for NkInt and A is a standard form convex cabling annulus.

The proof will be constructive and will be presented in Sections 7.2.2 and 7.2.3; we willbuild the Nk′

Int,(pG,q)inside of the original Nk

Int.

7.2. CABLING THEOREMS FOR POSITIVE INTERLOCKING SOLID TORI 111

Figure 7.9: Shown is ∂(NkInt ∪ A) with one component of its two dividing curves; the

boundary slope is −1/(pk + 1).

7.2.1 Weighting

Let X be the braid representing NkInt; so X is a braid representative for the knot type

K. Before undertaking the construction of cabling operations, we first use this subsectionto describe how to construct an interlocking steps configuration for q copies of X, givenNk

Int. We call this a weighting operation, and we will show that this construction yieldsan interlocking, positively twisting steps configuration for the link composed of q parallelcopies of X. The motivation for first looking at this weighting construction is that it is easyto understand, and provides a foundation upon which to explain the cabling constructionsin Sections 7.2.2 and 7.2.3.

Construction

The construction will be performed in the interior of NkInt. Specifically, inside each thickened

disc Di for NkInt we stack q new discs, labelled from greater z-value to lesser z-value as

Di1 , · · · , Diq . We then, for every λi-arc and corresponding θi-value for NkInt, associate an

ǫ-neighborhood [θi − ǫ, θi + ǫ] where (θi − ǫ) is the angle of the left edge of the top block ofthe (slightly thickened) (k + 1)-stack corresponding to λi, and (θi + ǫ) is the angle of theright edge of the bottom block of the (slightly thickened) (k +1)-stack corresponding to λi.We then introduce into this θ-interval q number of θ-values, which we label θi1 < · · · < θiq ,and where θi1 = (θi − ǫ) and θiq = (θi + ǫ). For both the original Di and the original θi forNk

Int, we are thinking of the ordering as given by traversing the positive braid X, and Di


will be the disc at the same z-level as the top of λi; this z-level will also correspond to thetop of the block Ri for Nk

Int. See Figure 7.10.

Figure 7.10: Shown is a local picture of the disc Di and λ-arc λi from the original NkInt; for q = 3,

the weighting discs Di1 ,Di2 ,Di3 and θ-values θi1 , θi2 , θi3 are indicated.

We now have a 1-1 correspondence between discs and θ-values, and we can describehow to place blocks across the support of the block Ri so as to obtain an interlocking,positively twisting, uniform steps configuration. The recipe is as follows: Let k′ = k/q. Fora given i, there will be a block with its top attached to Di1 , its bottom attached to D(i−1)1 ,its left edge at θi1 , and its right edge at θ(i+k′)1 . There will then be a block with its topattached to Di2 , its bottom attached to D(i−1)2 , its left edge at θ(i+k′)2 , and its right edgeat θ(i+2k′)2 . We continue across Ri, with the general block having its top attached to Dij ,its bottom attached to D(i−1)j

, its left edge at θ(i+(j−1)k′)j, and its right edge at θ(i+jk′)j

for 1 ≤ j ≤ q; in particular, the last block attached for Ri will have its right edge at θi+k,thus completing the blocks associated with Ri. A picture associated with this constructionis shown in Figure 7.11.

Now note that by construction we obtain a positively twisting steps configuration withstep size k′ = k/q; moreover, by construction it is immediate that λ(i+jk′)j

interlocks with

λ(i+jk′)j+1for 1 ≤ j < q. Then note that by the interlocking of Nk

Int we also have thatλ(i+qk′)q

interlocks with λ(i+1)1 (where he we have slightly abused notation, with λi+1 beingnow the next λ-arc in the θ-ordering). So this construction yields an interlocking, positivelytwisting, uniform steps configuration, and it deformation retracts onto q parallel copies ofX. We will denote this new steps configuration as Nk′

Int,q. Figure 7.12 shows a completed

example where NkInt represents a (2, 3) torus knot, k = 4, and the weighting is q = 2.

Weighting annuli

We now see how to attach q copies of a standard form convex annulus, denoted A1, · · · , Aq,to Nk′

Int,q so that each annulus has boundary components on two different solid torus com-


Figure 7.11: Shown is the weighting operation that can be performed, given an interlocking, posi-tively twisting uniform steps configuration Nk

Int. Specifically, outlined in light gray are the (thick-ened) discs and blocks of Nk

Int, where k = 6. For the weighting operation, q = 3, and so the groupsof 3 horizontal black lines represent Di1 , · · · ,Diq

, while the groups of 3 vertical dashed gray linesrepresent the θi1 , · · · , θiq

. The dark gray blocks are then placed according to the construction, andresult in q parallel copies of interlocking configurations, with step size k′ = 2.

Figure 7.12: Shown is a q = 2 weighting of NkInt representing the (2, 3) torus knot and where k = 4.


ponents of Nk′

Int,q, and so that one component of ∂N(Nk′

Int,q ∪ A1 ∪ · · · ∪ Aq) is ∂NkInt, and

the other component of ∂N(Nk′

Int,q ∪ A1 ∪ · · · ∪ Aq) bounds a standard Legendrian core of

NkInt. We will refer to the A1, · · · , Aq as weighting annuli. Essentially, we can form the

annuli A1, · · · , Aq in the following manner: Label the components of Nk′

Int,q as N1, · · · , Nq

from left to right, as viewed in the steps configuration diagram. We can then label G−+,i

and G+−,i as the appropriate graphs on ∂Ni, and for 1 ≤ i ≤ q − 1 we form Ai by simpleLegendrian isotopy of G−+,i to G+−,i+1 in an I-invariant neighborhood. See Figure 7.13,which shows the two such annuli for a weighted steps configuration where q = 3.

Figure 7.13: Shown are two of the three weighting annuli associated with this weighted stepsconfiguration where q = 3. The two different annuli have different angles of hatching to distinguishbetween them.

The final annulus, Aq, is also formed by Legendrian isotopy of G−+,q to G+−,1 in thecomplement of Nk′

Int,q; this isotopy cannot be performed in an I-invariant neighborhood, butit can be seen locally as a Legendrian isotopy of rectangular diagrams that come in front ofthe blocks as they appear in the steps configuration diagram. See Figure 7.14, which showsthe final annulus coming over the blocks for the case of q = 3. In this figure it is evidentthat one component (the exterior component) of ∂N(Nk′

Int,q ∪ A1 ∪ · · · ∪ Aq) is ∂NkInt.

Since these annuli represent Legendrian isotopies of their boundary components, theyare standard convex.

Standard neighborhood of a Legendrian

We now show that the interior component of ∂N(Nk′

Int,q ∪A1 ∪ · · · ∪Aq) bounds a standard

neighborhood of a Legendrian core curve for NkInt. To do this, we construct the standard


Figure 7.14: Shown are all three weighting annuli for a weighting of q = 3; the final annulus has45-degree hatching and appears in front of the gray blocks in the steps configuration.

neighborhood as a negatively twisting steps configuration; the construction can be seen bya local picture, an example of which is shown in Figure 7.15.

We explain both Figure 7.15 and the general construction. We first place (q − 1) discsthat interleave with the q discs associated to each original disc of the Nk

Int. We then showhow to place blocks down to form a steps configuration.

We first focus on the black blocks in Figure 7.15. Each of the black blocks is just tothe left of a λ-arc, and each black block’s z-support essentially covers the bottom k′ blocksof the (k′ + 1)-stack corresponding to that λ-arc. Moreover, for the black blocks, theircorresponding λ-arcs are for (k′ + 1)-stacks that compose the 2nd to (q − 1)-st strands ofthe braiding of the weighted X. Successive λ-arcs for the 2nd through (q − 1)-st strands ofthe braiding in Nk′

Int,q correspond to successive negatively twisting black blocks. The grayblocks then connect the bottom edges of blocks associated with the q − 1-st strand to thetop edges of blocks associated with the 2nd strand of the braiding of the weighted X. Theseare shown in the diagram as going down to the bottom of the grid, and appearing at the topagain; as we traverse the support of Nk

Int, these black and dark gray blocks thus deformationretract onto a core curve for Nk

Int. Since it is a negatively twisting steps configuration, itrepresents a standard neighborhood of a Legendrian knot.

7.2.2 Local positive G-framing cabling operations

We can now begin proving Theorem 7.2.1. We will have two cases, one where the cabling(pG, q) is positive in the G-framing, the other case being where q/pG < −(k + 1)/mk in the


Figure 7.15: Shown in dark gray and black blocks is the steps configuration for a standard neigh-borhood of a Legendrian, whose boundary is one component of ∂N(Nk′

Int,q ∪ A1 ∪ · · · ∪ Aq).


G-framing. In this subsection, we will prove the first case.

We will begin with a q-weighting of a positively interlocking uniform steps configurationNk

Int representing a braid X; we then show how to construct a positively interlocking uniformsteps configuration Nk′

Int,(pG,q)representing the cabling knot type K(pG,q) in the G-framing,

where pG ≥ 1 is an integer relatively prime to q. This cabling operation will be a localconstruction.

Before embarking on the construction, we show in Figure 7.16 an example of theN2

Int,(3G,2)representing the ((2, 3), (3G, 2)) iterated torus knot; it has been constructed inside

an N4Int representing the (2, 3) torus knot.

Figure 7.16: Shown is an interlocking steps configuration representing the (3G, 2)-cabling of the(2, 3) torus knot. The N2

Int,(3G,2) representing the (3G, 2)-cabling is built inside of the N4Int,(2,3)

representing the (2, 3) torus knot.


Construction

The construction proceeds as follows. We take the q-weighted Nk′

Int,q, but focus our attention

on a λ-arc of the original NkInt, which recall corresponds precisely to the θ-value of a λi1

in the weighted interlocking construction. We then choose δ > 0 so that θi + ǫ < θi + δ <θi+1 − ǫ. The half-plane Hθδ

intersects qk′ blocks associated to the strand of the braid Xcorresponding to the original λ-arc.

What we essentially do now is make a cut at Hθδand insert a steps configuration similar

to that for positive torus knots, which will result in the cabled braid X(pG,q). To this end,we first note that each of the qk′ blocks have their bottom edges attached to discs at theθδ-value. We insert p discs directly above each of these qk′ discs, and also insert pG discsabove the top-most disc associated with these qk′ blocks. See Figure 7.17, where pG = 3,and the groups of 3 discs are indicated by groups of 3 horizontal black lines.

Thus we have inserted (pGqk′ + pG) new discs. We now introduce (pGqk′ + pG) verticallines in a neighborhood of θδ. For purposes of the construction, however, it will be convenientto work with a (pGqk′ + pG + q) × (pGqk′ + pG + q) grid of discs and vertical lines, so weinclude the θ-values associated to λi1 through λiq that are contained in [λi − ǫ, λi + ǫ], andwe include the discs that are at the bottom of each of the q groups of k′ blocks.

We now show how to build the cabling interlocking steps configuration. We first pushthe left edges of the original qk′ blocks to the right of our (pGqk′ + p + q)× (pGqk′ + p + q)grid, keeping their vertical positions constant. The q blocks that complete the (k′+1)-stacksassociated with these qk′ blocks then have their vertical edges shifted up to the top discin their associated collection of pG discs. These q blocks will have their right edges on thefirst q vertical lines of the (pGqk′ + pG + q)× (pGqk′ + pG + q) grid. See Figure 7.17, whichshows the beginnings of a (3, 2)-cabling in the G-framing. For notational convenience wewill index the new discs as z0 > z1 > · · · > z(pGqk′+pG+q−1), ordered from top to bottom.The new θ-values will be labelled θ0 < θ1 < · · · < θ(pGqk′+pG+q−1).

We then begin in the top left of the (pGqk′ + pG + q)× (pGqk′ + pG + q) grid, and showhow to attach the right edges of new blocks to the final (pGqk′ + pG) vertical lines of ourgrid. The top block will become (relative to our (pGqk′ + pG + q) × (pGqk′ + pG + q) grid)a block of width qk′ and height pG (similar to the construction for positive torus knots),with its left edge at θ0. The blocks going backwards in the braiding will then have theirright edges q units behind the previous. This establishes a λ-arc at θ0, denoted as λ0 (thisis actually the same λ-arc from the weighting construction), and a (k′ + 1)-stack.

When we move down to the next group of k′ blocks, we again have the top block is a qk′

by pG block, with its left edge at θ1; the blocks going backwards in the braiding will thenhave their right edges q units behind the previous. This establishes a (k′ + 1)-stack with λ1

interlocking with λ0 (again, λ1 is the same λ-arc coming from the weighting construction).See Figure 7.18.

We continue this until we have finished the q groups of (k′ + 1) blocks on the left edgeof our grid, with interlocking λ-arcs.

We now show how to continue the construction across the grid by placing down single


Figure 7.17: Shown is the inserted discs needed for a positive (3, 2)-cabling as measured in theG-framing. The original blocks have been adjusted for the purposes of the cabling.

Figure 7.18: Shown are the beginnings of the interlocking steps configuration representing a (3, 2)-cabling in the G-framing.


blocks to make new (k′+1)-stacks. To see how to do this, we place a single block with its topedge at a z-level (pGk′+1) below the top edge of the last (k′+1)-stack, and with its left edgeat θq. This block will wrap around to the top and have its bottom edge attach to the top ofthe original (k′ + 1)-stack, and thus the left edge of the new block will represent a λθ1 for anew (k′ + 1)-stack. Note that by construction we still have λθq−1 and λθq

interlocking. Wecontinue this procedure iteratively: if a (k′ +1)-stack at λθi

has been completed, we place anew block down with its left edge at θi+1 and its top edge (pGk′+1) (cyclic) units down fromthe top edge of the last block placed. By construction, λθi

and λθi+1 will be interlocking.Moreover, since (pGk′+1) and (pGqk′+pG +q) are relatively prime, this iterative procedurewill result in every disc being identified with the top of a unique λ-arc, and the process willterminate once this is accomplished. At this termination point, the last block place will bethe one that has its top edge on the bottom disc of the (pGqk′ + pG + q)× (pGqk′ + pG + q)grid, and then wraps around to attach to the zpG−1-level. The λ-arc associated to this(k′ + 1)-stack will interlock with λi+1 from the weighting construction; thus the resultingNk′

Int,(pG,q)is interlocking.

Now within the cabling region, the G-longitude for the steps configuration has a con-stant slope of pG/q; i.e., over q units and up pG. Thus the braid resulting from the stepsconfiguration will embed on the original torus, and since pG and q are relatively prime, thesteps configuration will represent a (pG, q) cabling. See Figure 7.19.

Figure 7.19: Shown is the completed steps configuration for the local (3, 2)-cabling as measured inthe G-framing.

At this point the reader may wish to refer back to Figure 7.16, where the dark grayblocks represent the new blocks used for cabling.


Cabling annulus and standard neighborhood of a Legendrian

In this section we first explain how to construct the cabling annulus A that results in onecomponent of ∂N(A∪Nk′

Int,(pG,q)) being ∂Nk

Int. Essentially the annulus is again obtained by

a Legendrian isotopy of G+− to G−+ on ∂Nk′

Int,(pG,q)as in the case of the weighting annuli;

in the cabling case, however, there is only one component. The way this occurs is thataway from the cabling region, the annulus looks just like as in the case of the weightingannuli, but as the annulus proceeds through the cabling region, the distinct components ofthe weighting annuli are joined together to create one knotted cabling annulus. Figure 7.20shows one portion of the annulus as it proceeds through the cabling region. Specifically, asit comes in from the left it looks like part of a weighting annulus that passes in front of theblocks, but after entering the cabling region it is then obtained by Legendrian isotopy ofG+− to G−+ in an I-invariant neighborhood. In the transitional region, Legendrian flipsprovide part of the isotopy.

Figure 7.20: Shown is one portion of the cabling annulus as it proceeds through the cabling region.

Figure 7.21 shows another portion of the annulus as it proceeds through the cablingregion. Specifically, as it comes in from the right it looks like part of a weighting annulusthat passes in front of the blocks, but after entering the cabling region it is then obtainedby Legendrian isotopy of G+− to G−+ in an I-invariant neighborhood.

Figure 7.22 then shows the two portions of the cabling annulus put together.

Figures 7.20, 7.21, and 7.22 are for the case of a (3, 2)-cabling. For q > 2, there willbe other portions of the cabling annulus that pass through the cabling region simply asLegendrian isotopies of G+− to G−+ in an I-invariant neighborhood.

It is evident that one component of ∂N(A ∪Nk′

Int,(pG,q)) will in fact be ∂Nk

Int, and from

this construction one can more directly see that the steps presentation Nk′

Int,(pG,q)in fact


Figure 7.21: Shown is another portion of the cabling annulus as it proceeds through the cablingregion; away from the cabling region the annulus simply looks like one of the original weightingannuli.

Figure 7.22: Shown is the total cabling annulus.


represents a (pG, q) cabling of X in the G-framing.

We can now see how the standard neighborhood of a Legendrian bounded by the othercomponent of ∂N(A ∪ Nk′

Int,(pG,q)) can be constructed similar to the weighted case. Specif-

ically, we again place discs that interleave with the existing q or (q + pG) discs associatedto each original disc of Nk

Int. Then away from the new blocks introduced by the cabling,the blocks composing the standard neighborhood of a Legendrian are precisely the same asin the weighting case; in Figure 7.23 these are indicated by the long thin dark gray blocks.Within the cabling, we simply place blocks to the left of each λ-arc as in the weighting; inthis case, blocks associated to consecutive λ-arcs in the θ-ordering will have their bottomand top edges, respectively, on the same disc, and thus the resulting steps configurationwill be negatively twisting and represent a core of Nk

Int, and hence will be the standardneighborhood of a Legendrian.

Figure 7.23: Shown in hatching is the cabling annulus A which results in one component of ∂N(A∪Nk′

Int,(pG,q)) being ∂NkInt. Also shown, with heavy gray discs and dark gray and black blocks, is the

standard neighborhood of a Legendrian bounded by the other component of ∂N(A ∪ Nk′

Int,(pG,q)).

Calculating boundary slopes

If there were originally mk number of blocks in the original NkInt, a count shows that there

are now qmk + pGqk′ + pG number of blocks for Nk′

Int,(pG,q). Thus in the G-framing for

∂Nk′

Int,(pG,q), the boundary slope will be −(k′ + 1)/(qmk + pGqk′ + pG). Furthermore, note

that for this positive cabling, the G-framing on ∂Nk′

Int,(pG,q)corresponds to the C′ framing

coming from the cabling annulus.

Also, the boundary slope for the standard neighborhood of the Legendrian knot, as


measured in the G-framing for ∂NkInt, will be −1/(mk + pGk′).

7.2.3 Global negative G-framing cabling operations

In this section we prove Theorem 7.2.1 for the case where q/pG < −(k + 1)/mk.

So given a positively twisting interlocking steps configuration NkInt representing a braid

X, and an integer q such that qk′ = k for an integer k′, we show how to construct a positivelytwisting interlocking steps configuration Nk′

Int,(pG,q)which represents a (pG, q) cabling (in the

G-framing) of the braid X. However, in this section pG will be a negative integer co-primeto q, and there will be restrictions on what values pG can attain. In particular, pG mustsatisfy |pG|/q < mk/(k + 1), where mk is the number of blocks in Nk

Int.

Before embarking on the construction, we show in Figure 7.24 an example of theN1

Int,(29G,3)representing the ((3, 4), (29G, 3)) iterated torus knot; it has been constructed

inside an N3Int representing the (3, 4) torus knot.

Construction

The construction, as opposed to the local one for positive cablings, will be a global con-struction over the total braid support of Nk

Int. We will refer the reader to Figures 7.25,7.26, 7.27, and 7.28 for examples of different aspects of the construction.

We begin locally, along a portion of the steps configuration which represents a singlestrand of Nk

Int. For this construction we begin by placing groups of k′ blocks that haveheights and widths that are just a small value ǫ less than the heights and widths of theoriginal blocks for Nk

Int. The pattern is as follows: As we move along a strand of the braidrepresented by the steps configuration for Nk

Int, we place q groups of k′ blocks in successionso that they almost exactly correspond to the original blocks of Nk

Int, but do not identify thetop-most and bottom-most edges of successive groups. Rather, we place two discs withinthe Nk

Int at that z-level, and attach the top-most and bottom-most of successive k′-stackson two different discs. We also place discs at the z-levels of top edges of blocks within eachstack. After laying down q groups, we skip an original block from Nk

Int and then repeatlaying down another group of q k′-stacks. See the light gray blocks in Figure 7.25, whereq = 4 and k′ = 2.

After placing down at least 2 of these q groups of k′-stacks, there will be left edgesof new blocks that correspond to the θ-levels of λ-arcs of the original Nk

Int, and for such agiven left edge there will be a unique right edge of a new block that corresponds to the sameθ-value. We then place a block that connects these two blocks and continues the step size ofk′; this new block goes over existing blocks as we view them in the steps configuration. Seethe dark gray blocks in Figure 7.25. From the figure one can see that this pattern resultsin a full negative twist of the q strands along the support of the original braid X, and isresulting in negative twisting with respect to the G-framing from Nk

Int (this will becomeclearer in the next subsection when we examine the cabling annulus).


Figure 7.24: Shown is N1Int,(29G,3) representing the ((3, 4), (29G, 3)) iterated torus knot; it has been

constructed inside an N3Int representing the (3, 4) torus knot.


Figure 7.25: Shown is part of a negative cabling where q = 4 and k′ = 2.

Notice that the λ-arcs for the new developing Nk′

Int,(pG,q)are interlocking, as they cor-

respond to the original λ-arcs from NkInt. One can see this in Figure 7.25 for q = 4 and

k′ = 2, or in Figure 7.26 for q = 2 and k′ = 2.

For pG satisfying |pG|/q < mk/(k + 1), we cease this twisting construction once thereare |pG| longer blocks which pass over the k′-stacks (i.e., the longer dark gray blocks as inFigure 7.25). We then fill in the rest of the support of Nk

Int with q-weighting and step sizek′, as defined above and shown in Figures 7.27 and 7.28. Note that there will always besome weighting portion of such a negative cabling construction, since pG and q are relativelyprime.

The reason that we require |pG|/q < mk/(k + 1) can be seen as follows: If we imagine aq-weighting of Nk

Int superimposed on Nk′

Int,(pG,q, then each of the negative twists associated

to the |pG| blocks will account for (k + 1) of the weighting blocks. Thus we must have thatqmk−|pG|(k+1) > 0, which is true if and only if |pG|/q < mk/(k+1). We will thus call thepositive integer qmk + pG(k + 1) the number of weighting blocks, as it represents the extraweighting blocks needed to complete the cabling. The total number of blocks in Nk′

Int,(pG,q)

will thus be qmk + pG(k + 1) − pG(k′ + 1).

Cabling annulus and standard neighborhood of Legendrian

Within the weighting portion of the negative cabling construction, the cabling annulus isidentical to that for the weighting constructon, namely a standard convex annulus formedby Legendrian isotopy of subarcs of G+− to G−+. However, in the negatively twistingportions of the construction formed by negative full twists of the q strands, the cabling


Figure 7.26: Shown is a portion of a negative cabling for k′ = 2 and q = 2.

Figure 7.27: Shown is a portion of a negative cabling with k′ = 1 and q = 3, where it transitionsfrom twisting to weighting.


Figure 7.28: Shown is a portion of a negative cabling with k′ = 1 and q = 3 where it transitionsfrom weighting back to twisting.

annulus is formed by Legendrian isotopy of G++ to G−−. See Figure 7.29, in which theisotopy of different subarcs of G++ to G−− are specified with different hatching.

In Figure 7.30 we then show the transition of the cabling annulus from the twistingportion to the weighting portion. Note that one boundary component of ∂N(Nk′

Int,(pG,q)∪A)

is indeed the original ∂NkInt.

To describe the construction of the standard neighborhood of the Legendrian core ofNk

Int which is bounded by the inside component of ∂N(Nk′

Int,(pG,q)), we will refer to Figures

7.31, 7.32, 7.33, and 7.34.

We first (as usual) place discs which interleave within the collections of either 2 (twistingportion) or q (weighting portion) discs associated with the original discs of Nk

Int. Wethen describe how to place blocks so as to obtain a block-disc presentation for a standardneighborhood of a Legendrian core for Nk

Int.

We begin within the negatively twisting portions of the steps configuration for Nk′

Int,(pG,q).

If we look at consecutive discs in the braid-ordering, we connect these by blocks whose leftedge is at the θ-level of the λ-arc whose top is at the original disc for Nk

Int at the upperz-level, and whose right edge is at the θ-level associated to the λ-arc whose bottom is atthe original disc for Nk

Int at the bottom z-level. See the black blocks in Figures 7.31, 7.33,and 7.34. Note that the resulting steps configuration locally looks as though it is positivelytwisting, and hence not a standard neighborhood; however, we will see that the weightingportion will correct for this.

Now the placement of blocks in the weighted portion is precisely what we had before;


Figure 7.29: Shown is a portion of the negative cabling with k′ = 1 and q = 3, along with thecabling annulus.

Figure 7.30: Shown is the cabling annulus in the region of transition between weighting and twisting.


Figure 7.31: Shown is a negative twisting portion of the negative cabling construction where q = 3and k′ = 1, along with the standard neighborhood of the Legendrian core in black.

see the top portion of Figure 7.32. This forces G++ and G−− to have one single longitu-dinal component after Giroux Elimination; this shows that in fact the resulting block-discconfiguration will represent a standard neighborhood of a Legendrian. Figure 7.32 alsoshows the transition between the negatively twisting and weighting portions of the stan-dard neighborhood; notice that the resulting block-disc configuration is not homogeneouslytwisting.

Calculating boundary slopes

Suppose that NkInt is our original positively twisting interlocking steps configuration, and

then we perform a cabling operation to obtain a new Nk′

Int,(pG,q)with pG < 0. Then the

change of basis from the ∞′ framing to the G-framing is given by µG = µ′ + pGλ for a(µ, λ) curve; this is because the algebraic intersection of the ∞′ longitude with the G-longitude is pG, i.e., one negative intersection for each long vertical block representing asingle negative twist. Furthermore, in the ∞′-framing, the intersection boundary slope isslope(Γ) = −(k′ + 1)/[qmk + pG(k + 1)] = −(k′ + 1)/[# of weighting blocks]; this is sinceaway from the weighting blocks, G++ runs parallel to the ∞′-longitude. Also, the totalnumber of blocks is qmk + pG(k + 1) − pG(k′ + 1). Moreover, the boundary slope for thestandard neighborhood of a Legendrian knot, as measured in the G-framing for ∂Nk

Int, is−1/(mk + pGk′).


Figure 7.32: Shown is the transition between the negatively twisting and weighting portions of thenegative cabling construction for q = 3 and k′ = 1, along with the standard neighborhood of theLegendrian core in black. Notice that the resulting solid torus is in fact a standard neighborhoodafter Giroux Elimination.

Figure 7.33: Shown is a negatively twisting portion of the negative cabling construction with q = 2and k′ = 2, along with the standard neighborhood of a Legendrian core in black.


Figure 7.34: Shown is a negatively twisting portion of the negative cabling construction with q = 4and k′ = 2, with the standard neighborhood of a Legendrian core in black.

7.3 Interlocking non-thickenables for iterated torus knots

We now show that every non-thickenable solid torus in the class of iterated torus knots canbe represented by an interlocking, positively twisting steps configuration:

Theorem 7.3.1 Let Kr = ((P1, q1), · · · , (Pi, qi), · · · , (Pr, qr)) be an iterated torus knot forwhich Pi > 0 for all i. Suppose Nk

r is a non-trivial non-thickenable solid torus representingKr. Then Nk

r is represented by a positively twisting interlocking steps configuration withstep size k.

Proof. We have completed the base case of positive torus knots. We therefore per-form induction on the iteration index r, and we may assume inductively that if Kr =((P1, q1), ..., (Pr, qr)) is an iterated torus knot where Pi > 0 for all i, then all of its non-thickenable solid tori can be represented as interlocking, positively twisting steps configu-rations, i.e., Nk

r = NkInt,r. Moreover, if r = 1, they are precisely the Nk

Int,1 established in

the previous subsection, and if r > 1, then NkInt,r was obtained from Nk′′

Int,r−1 by one of the

above cabling operations, either for pGr > 0 or pG

r < 0. Furthermore, the following is true:

• If r = 1, or if pGr > 0 for r > 1, then ∞′

r = ∞Gr ; also, the total number of blocks in

the steps configuration is Ark + Br. This calculation for the number of blocks followsfrom the calculation at the end of Section 7.2.2 as well as the recursive identities forAr and Br from Chapter 2.

7.3. INTERLOCKING NON-THICKENABLES FOR ITERATED TORUS KNOTS 133

• If pGr < 0 for r > 1, then ∞′

r is a (pGr , 1)-cabling in the G-framing, so that for a simple

closed curve (µ, λ) on ∂NkInt,r in the ∞′ framing, we have µG = µ′ + pG

r λ. Also, the

total number of blocks is Ark + Br − pGr (k + 1). This calculation for the number of

blocks follows from the calculation at the end of Section 7.2.3.

Now let Pr+1 > 0, and let Nk′

r+1 be a non-thickenable representing Kr+1. We will show

that Nk′

r+1 can be represented by an Nk′

Int,r+1, and that the above inductive hypothesis is

preserved. First note that we must have Nk′

r+1 ⊂ NkInt,r, and we know that in the C′ framing

that slope(Γr) = −(k + 1)/(Ark + Br). We also know that qr+1k′ = k, and that either

pr+1 > 0 or qr+1/pr+1 < −(k + 1)/(Ark + Br).

We first verify the construction as well as the appropriate boundary slopes. There aretwo cases, and then two subcases within each case.

Case 1: r = 1 or pGr > 0 for r > 1. Then the ∞′

r framing is the G-framing,so pr+1 = pG

r+1. Thus, if pr+1 > 0, we can form a positively twisting interlocking stepsconfiguration representing the (pr+1, qr+1) cabling using the positive cabling operation; andif pr+1 < 0, we can form a positively twisting interlocking steps configuration representingthe (pr+1, qr+1) cabling using the negative cabling operation, since qr+1/pr+1 < −(k +1)/(Ark + Br) = −(k + 1)/[# of blocks].

We first take the case pr+1 > 0. Then the total number of blocks is qr+1(Arqr+1k′ +

Br) + pr+1(qr+1k′ + 1) = Ar+1k

′ + Br+1. Thus the intersection boundary slope is −(k′ +1)/(Ar+1k

′ + Br+1).

Then, if pr+1 < 0, then the number of weighting blocks is qr+1(Arqr+1k′ + Br) +

pr+1(qr+1k′+1) = Ar+1k

′+Br+1. Thus the intersection boundary slope is −(k′+1)/(Ar+1k′+

Br+1). Furthermore, the total number of blocks is Ar+1k′ + Br+1 − pG

r+1(k′ + 1).

Case 2: pGr < 0. Then ∞′

r is a (pGr , 1)-cabling in the G-framing, so that for a simple

closed curve (µ, λ) on ∂NkInt,r in the ∞′ framing, we have µG = µ′ + pG

r λ. Also, the total

number of blocks is Ark + Br − pGr (k + 1).

Now in the C′ framing, we know that qr+1/pr+1 < −(k+1)/(Ark+Br). But this impliesthat the algebraic intersection of (pr+1, qr+1) with (−(Ark+Br), k+1) is positive. Thus, inthe G-framing, the algebraic intersection of (pG

r+1, qr+1) with (−[(Ark+Br)+pGr (k+1)], k+1)

is positive. But this implies that pGr+1/qr+1 > (Ark + Br − (pG

r (k + 1))/ − (k + 1), whichmeans that either pG

r+1 > 0 or qr+1/pGr+1 < −(k + 1)/[total # of blocks]. As a consequence,

if pGr+1 > 0, we can form a positively twisting interlocking steps configuration representing

the (pr+1, qr+1) cabling using the positive cabling operation; and if pGr+1 < 0, we can form

a positively twisting interlocking steps configuration representing the (pr+1, qr+1) cablingusing the negative cabling operation.

To calculate boundary slopes, if pGr+1 > 0, we have that the total number of blocks is

qr+1[Ark + Br − pGr k − pG

r ] + (pr+1 + pGr qr+1)qr+1k

′ + pGr+1, which after simplifying equals

Ar+1k′ + Br+1. Thus the intersection boundary slope is −(k′ + 1)/(Ar+1k

′ + Br+1).


If pGr+1 < 0, we have that the number of weighting blocks is qr+1[Ark + Br − pG

r (k +1)] + pG

r+1(k′qr+1 + 1), which after simplifying equals Ar+1k

′ + Br+1. Thus the intersectionboundary slope is −(k′ + 1)/(Ar+1k

′ + Br+1), and the total number of blocks is Ar+1k′ +

Br+1 − pGr+1(k

′ + 1).

It now remains to note that by previous subsections, we know that the Nk′

Int,r+1 have

complements that decompose precisely into that of the Nk′

r+1. Thus the Nk′

Int,r+1 in fact arenon-thickenable. �

7.4 Legendrian and transversal MTWS for ((2, 3), (1, 2))

We conclude this chapter by establishing a knot type-specific Legendrian and transversalMarkov theorem without stabilization for the (1, 2)-cabling of the (2, 3) torus knot, thusexpanding upon the classification stated in Theorem 5.2.1. The strategy will be similar tothat for proving the Legendrian MTWS associated with the (2, 3)-cabling of the (2, 3) torusknot in [32]; however, for this example, elementary negative flypes are not the moves thatallow us to jump between isotopy classes at the same values of tb and r, but rather a movewhich we call a cyclic move, following [5]. This cyclic move will be composed of a singlestabilization, followed by a sequence of exchange moves, followed by a destabilization, allof which can be seen on the cabling torus. We will thus prove the following theorem andcorollary (the reader may want to refer back to Theorem 5.2.1 to review the notation there).

Theorem 7.4.1 Let K ′ be the (1, 2)-cable of a (2, 3)-torus knot. Then Legendrian positiveand negative destabilizations (modulo Legendrian isotopy), along with cyclic moves, aresufficient to take a Legendrian representative of K ′ to L± (the two tb representatives onthickenable solid tori). In particular, we have the following:

1. L− is related by a cyclic move to Lnt− .

2. L+ is related by a cyclic move to Lnt+ .

Using transversal push-offs, we have the following theorem:

Theorem 7.4.2 Let K ′ be the (1, 2)-cable of a (2, 3)-torus knot. Then negative braid desta-bilizations, along with cyclic moves, are sufficient to take a transversal representative of K ′

to a representative at maximal self-linking number, modulo transversal isotopy. In particu-lar, T+(L+) is related by a cyclic move to T+(Lnt

+ ).

It suffices to prove Theorem 7.4.2 by actually exhibiting the cyclic move on the surfaceof the cabling torus, and thus as a sequence of braided rectangular diagrams; Theorem 7.4.1then follows since braided Legendrian rectangular diagrams are identical to their positivetransverse push-offs. We show the sequence of moves first in the braid foliation in Fig-ures 7.35 through 7.40; note that two non-standard changes of fibration permit the second

7.4. LEGENDRIAN AND TRANSVERSAL MTWS FOR ((2, 3), (1, 2)) 135

destabilization in the cyclic move. We then show the sequence of moves as braided rectan-gular diagrams superimposed on the interlocking solid torus in Figures 7.41 through 7.46.These rectangular diagrams represent both L+ (in the Legendrian case), and T+(L+) (inthe transversal case), and are thus sufficient to prov both theorems. The resulting solidtorus in Figure 7.46 is indeed a thickenable solid torus, as evidenced in the final Figure7.47.

Figure 7.35: Shown is T+(Lnt+ ) embedded in the braid foliation of ∂N1

1 .


Figure 7.36: Shown is the first stabilization of T+(Lnt+ ).

Figure 7.37: Shown is the first non-standard change of fibration, which is possible due to the firststabilization.


Figure 7.38: Shown is the second non-standard change of fibration, made possible by the firstnon-standard change of fibration.

Figure 7.39: Shown is the sequence of 3 successive exchange moves in the cyclic move.


Figure 7.40: Shown is the destabilization resulting in T+(L+). It is now on the boundary of athickenable solid torus.

Figure 7.41: Shown is T+(Lnt+ ) embedded in the boundary of the interlocking solid torus.


Figure 7.42: Shown is the first stabilization on the boundary of the interlocking solid torus.

Figure 7.43: Shown is the block-disc presentation following the two non-standard changes of fibra-tion.


Figure 7.44: Shown is the destabilizing disc following the 3 exchange moves.

Figure 7.45: Shown is the destabilized T+(L+).


Figure 7.46: Shown is the thickenable solid torus containing T+(L+).

Figure 7.47: Shown in (a) is the solid torus with boundary slope −1/5 which thins to thesolid torus in (b) having boundary slope −2/11, thus demonstrating that the solid torus in(b) indeed is thickenable.


Chapter 8

Open questions and futuredirections

In this chapter we discuss open questions and future directions, many of which overlap andare very much related.

8.1 Positively twisting interlocking solid tori

8.1.1 General constructions

In Chapter 7, we showed how to construct new positively twisting interlocking solid torifrom old positively twisting interlocking solid tori, and this resulted in solid tori that rep-resented cabled knot types. However, it seems evident that such constructions need not beconstrained to the context of cabling. For example, we could take a weighted constructionfor a torus knot and perform a positively twisting operation similar to that for the positivecabling operation, but one which yields a positively interlocking solid torus representing atwisted T-link [4]. This yields the following question:

Question 8.1.1 What knots other than cabled knot types can be represented by positivelytwisting interlocking solid tori?

A related question is the following:

Question 8.1.2 Are there a finite set of moves that can be performed to take one positivelytwisting interlocking solid torus to another (even across different knot types)?

8.1.2 Sequences of interlocking solid tori

In all of our examples of positively twisting interlocking solid tori representing iterated torusknots, there existed an infinite sequence of such solid tori for that knot type, representinga sequence of non-thickenable solid tori that resulted in the knot type χ-failing the UTP.

143

144 CHAPTER 8. OPEN QUESTIONS AND FUTURE DIRECTIONS

The intersection boundary slopes of these solid tori were −(k + 1)/χ(K), and k → ∞ wasrealized by a corresponding unbounded increase in step size for the interlocking solid tori.This yields the following question:

Question 8.1.3 Suppose N is a positively twisting interlocking solid torus representing aknot type K, with step size k. Then for k′ > k, does there always exist an interlocking N ′

representing K with step size k′?

This seems reasonable, as there is no obstruction immediately evident that would preventone from arbitrarily increasing step size.

8.1.3 Boundary slopes of interlocking solid tori

In all of our examples for interlocking solid tori representing iterated torus knots, the inter-section boundary slopes were −(k+1)/χ(K); in particular, the intersection with a preferredlongitude was always −χ(K). This yields the following question:

Question 8.1.4 Suppose N is a positively twisting interlocking solid torus representing K.Then is it the case that the intersection boundary slope is −(k + 1)/χ(K)? If not, is it thecase that the intersection boundary slope is (k + 1)/m for some fixed positive integer m?

8.1.4 Interlocking solid tori as non-thickenables

In the class of iterated torus knots, every non-thickenable could be represented as an inter-locking solid torus. It is thus reasonable to ask the following questions:

Question 8.1.5 Suppose N representing K is a non-trivial non-thickenable solid torus. IsN represented by a positively twisting interlocking solid torus?

Question 8.1.6 Suppose N is a positively twisting interlocking solid torus. Is N then anon-thickenable?

The proofs that established this for iterated torus knots very much relied on the factthat iterated torus knots are fibered with periodic monodromy, and were based in convexsurface theory. It is therefore likely that new ideas connecting convex surface theory andbraid foliations will be needed to understand the more general relationship of interlockingsolid tori and non-thickenables.

8.1.5 Interlocking solid tori for links

The weighting operation discussed in Section 7.2.1 suggests that we need not restrict ourstudy to interlocking solid tori representing knots, but rather should be able to extend theabove questions for knots to similar questions for links.

8.2. THE LOWER WIDTH 145

8.2 The lower width

8.2.1 Lower bound for the lower width

As mentioned above, in the class of iterated torus knots, the intersection boundary slopesof all non-thickenables are positive; hence the lower width is greater than or equal to zero.This yields the following question:

Question 8.2.1 If K is a knot type that fails the UTP due to the presence of non-thickenables,is it always the case that lw(K) ≥ 0?

8.2.2 Lower width of zero

Since iterated torus knots that fail the UTP actually χ-fail the UTP, they all have lowerwidths that equal zero. We thus have the following question:

Question 8.2.2 If K is a knot type that fails the UTP due to the presence of non-thickenables,is it always the case that lw(K) = 0?

If in fact non-thickenables can be represented as interlocking solid tori, and the stepsize of interlocking solid tori can be increased without bound, and the boundary slopes arepositive, the answer to the above question would be yes.

8.2.3 Integral lower width

A related, but weaker, question to the one above is the following:

Question 8.2.3 Are there knots K for which lw /∈ Z?

8.2.4 Negative cablings and the UTP

A positive answer to Question 8.2.1 would mean that if K fails the UTP due to the presenceof non-thickenables, then negative cablings of K would satisfy the UTP. We thus have thefollowing question:

Question 8.2.4 If K fails the UTP, and P < 0, is it the case that K(P,q) satisfies theUTP?

8.3 The UTP and fibered knots

8.3.1 Strongly quasipositive knots

Hedden has shown that for general fibered knots K in S3, ξK∼= ξstd precisely when K is

a fibered strongly quasipositive knot [27]; he also shows that for these knots, the maximalself-linking number is sl(K) = −χ(K). Furthermore, from the work of Etnyre and Van


Horn-Morris [19], we know that for fibered knots K in S3 that support the standard contactstructure there is a unique transversal isotopy class at sl. In the present work, all of theseideas are brought to bear on the class of iterated torus knots, and this motivates the followingquestion concerning general fibered knots:

Question 8.3.1 Let K be a fibered knot in S3. Is it the case K fails the UTP if and onlyif ξK

∼= ξstd, and hence if and only if K is fibered strongly quasipositive? Moreover, is it thecase that if a topologically non-trivial fibered knot K fails the UTP, then it supports cablingsthat are transversally non-simple?

We also ask the following question of non-fibered knots in S3:

Question 8.3.2 If K is a non-fibered strongly quasipositive knot, does K fail the UTP andsupport transversally non-simple cablings?

It seems reasonable that these questions can be at least experimentally investigated bytrying to construct interlocking solid tori representing both fibered strongly quasipositiveknots, as well as non-fibered strongly quasipositive knots, as strongly quasipositive knotshave convenient braid presentations as the boundary of surfaces that are simply discs bandedtogether by bands with positive half-twists.

8.3.2 The figure eight knot

The figure eight knot is perhaps the simplest knot known which does not support thestandard contact structure. Thus it represents a natural first test case for the above questionof whether knots which fail to support the standard contact structure then satisfy the UTP:

Question 8.3.3 Is it the case that the figure eight knot satisfies the UTP?

Etnyre and Honda have shown that the figure eight knot is Legendrian simple in [18],and in so doing have heavily analyzed the figure eight knot complement from the pointof view of convex surface theory; a careful study of their proofs would thus be a naturalstarting point for attacking the above question.

8.3.3 Open book decompositions of contact 3-manifolds

The above considerations motivate the following question:

Question 8.3.4 Suppose (M, ξ) is a tight contact 3-manifold, and K is a fibered knot type.Then can a suitable UTPM be defined so that K either satisfies or fails the UTP? If so,is it the case that K fails the UTP if and only if ξK

∼= ξ? And does such a K supporttransversally non-simple cablings (if the classical invariants can be suitably defined)?

8.4. FAILING THE UTP IN S3 147

8.4 Failing the UTP in S3

At the moment, the knots which we know fail the UTP break down into two sets: thosefor which w 6= tb, and those that fail the UTP due to the presence of non-thickenables. Atthe moment, the only element known in the first set is the unknot, and the only elementsin the second set are iterated torus knots with Pi > 0 for all i. This yields the followingquestions, which we separate into subsections.

8.4.1 tb > w

The main question here is if any other knot fails the UTP in the way the unknot does.Specifically, we have the following question:

Question 8.4.1 For what knots K is it the case that tb(K) > w(K)?

8.4.2 χ-failing the UTP

The main question here is how frequently do knots that fail the UTP actually do so byχ-failing the UTP. We thus have the following:

Question 8.4.2 What knots K fail the UTP by χ-failing the UTP?

8.4.3 Non-integral contact width

The main question here is whether there is a knot for which w(K) /∈ Z:

Question 8.4.3 Are there knots K for which w(K) /∈ Z?

8.4.4 Non-thickenables even though tb = w = lw

The main question here is whether there can be knots that fail the UTP even thoughtb = w = lw. We thus have the following question:

Question 8.4.4 Are there knots K with tb(K) = w(K) = lw(K) but that fail the UTP,due to a non-thickenable with minimal intersection boundary slope n/(ntb) where n > 1;i.e., the boundary slope is 1/tb, but the minimal number of dividing curves is greater than2?

8.5 Legendrian classification of cabled knot types

8.5.1 Classification of iterated torus knots

The goal here is to provide a complete Legendrian and transversal classification of all isotopyclasses for all iterated torus knots in S3. This would significantly expand the classes of knotsfor which we have complete Legendrian classifications, and is currently joint work with


Tosun, who has independently made progress on the question of which cablings of a non-UTP knot type are non-simple [44]. Our goal is to do the following: Given an iterated torusknot type Kr = ((p1, q1), ..., (pr, qr)), determine from this r-tuple of ordered pairs whetheror not Kr is transversally or Legendrian simple, and provide a complete classification ofall the Legendrian isotopy classes for that knot type. This would be the first Legendrianclassification for such a large historically self-contained class of knots.

8.5.2 Classification of cablings of fibered knots

Note that answers to the above questions may actually combine to make significant progresstoward understanding the extent to which cablings of fibered knot types can be transversallynon-simple in S3. Such a classification of cablings of fibered knots would make significantprogress toward understanding the interplay between open book decompositions and non-simplicity.

8.6 Other applications of the UTP

8.6.1 Hyperbolic knots and the UTP

There are no known substantive connections between contact geometry and hyperbolicgeometry; however, questions related to strongly quasipositive knots and recent work ofBirman and Kofman on Lorenz knots suggests a possible line of inquiry [4]. They havefound that many hyperbolic knots with small hyperbolic volume are Lorenz. Lorenz knotsare a subset of strongly quasipositive knots, and thus a question for future research presentsitself: Are there relationships between hyperbolic knots that fail the UTP and the volume oftheir knot complements? This is certainly a very open question, but is worth investigatingsince any connection found would be interesting.

8.6.2 Non-transversal global isotopies

A fourth line of work is to continue a more detailed analysis of the global isotopies re-quired to move between transversal isotopy classes at the same self-linking number. Onecan first consider this question in the context of iterated torus knots; the goal is to takethe Legendrian classification for non-simple iterated torus knots, and then determine theisotopies required to jump between isotopy classes at the same classical invariants. This isan extension of the example given above for the (1, 2)-cabling of the (2, 3) torus knot, anda similar strategy should be able to be employed, namely using braid foliation techniquesto analyze the Legendrian classification that comes out of convex surface theory.

One can also consider the question of global isotopies in the context of general knottypes. This is joint work with Menasco, based on recent work of Matsuda in the topologicalbraid setting [33]. Matsuda has developed a setting in which to begin to understand pre-cisely what global isotopies are required to move between braid representatives of the sametopological knot type; our goal is to be able to study these isotopies to determine which

8.6. OTHER APPLICATIONS OF THE UTP 149

yield isotopies that allow for non-transversal jumps between transversal isotopy classes.This analysis would begin to provide a more complete braid-theoretic framework withinwhich to understand potential classifying invariants for transversal knots.


Bibliography

[1] J.W. Alexander. A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. USA Vol.9 (1923), 23–25.

[2] D. Bennequin, Entrelacements et equations de Pfaff, Asterisque No. 107-108 (1983),87–161.

[3] J. Birman and E. Finkelstein, Studying surfaces via closed briads, J. Knot Theory Ram-ifications Vol. 7 No. 3 (1998), 267–334.

[4] J. Birman and I. Kofman, A new twist on Lorenz links, J. Topol. No. 2 (2009), 227–248.

[5] J. Birman and W. Menasco, Stabilization in the braid groups I: MTWS, Geom. Topol.Vol.10 (2006), 413–540.

[6] J. Birman and W. Menasco, Stabilization in the braid groups II: transversal simplicityof transverse knots, Geom. Topol. Vol. 10 (2006), 1425–1452.

[7] J. Birman and W. Menasco, On Markov’s Theorem, J. Knot Theorey Ramifications Vol.11 No. 3 (2002), 295–310.

[8] J. Birman and N. Wrinkle, On transversally simple knots, J. Differential Geom. Vol. 55No. 2 (2000), 325–354.

[9] Y. Eliashberg, Classification of overtwisted contact structures on 3-manifolds, Invent.Math. Vol. 98 (1989), 623–647.

[10] Y. Eliashberg, Contact 3-manifolds, twenty years since J Martinet’s work, Ann. Inst.Fourier Vol. 42 (1992), 165–192.

[11] Y. Eliashberg and M. Fraser, Topologically trivial Legendrian knots, J. SymplecticGeom. Vol. 7 No. 2 (2009), 77–127.

[12] J. Epstein, D. Fuchs, and M. Meyer, Chekanov-Eliashberg invariants and transverseapproximations of Legendrian knots, Pacific J. Math Vol. 201 No. 1 (2001), 89–106.

[13] J. Etnyre, Tight contact structures on lens spaces, Commun. Contemp. Math. Vol. 2No. 4 (2000), 559–577.

151

152 BIBLIOGRAPHY

[14] J. Etnyre, Erratum to Tight contact structures on lens spaces, Commun. Contemp.Math. Vol. 3 No. 4 (2001), 649–652.

[15] J. Etnyre, Legendrian and Transversal Knots, in Handbook of Knot Theory, Elsevier(2005), 105–185.

[16] J. Etnyre and R. Ghrist, Contact topology and hydrodynamics III: Knotted flowlines,Trans. Amer. Math. Soc. No. 352 (2000), 5781–5794.

[17] J. Etnyre and K. Honda, Cabling and transverse simplicity, Ann. of Math. (2) Vol. 162No.3 (2005), 1305–1333.

[18] J. Etnyre and K. Honda, Knots and Contact Geometry I: Torus Knots and the FigureEight Knot, J. Symplectic Geom. Vol. 1 No. 1 (2001), 63–120.

[19] J. Etnyre and J. Van Horn-Morris, Fibered transverse knots and the Bennequin bound,e-print at arXiv:0803.0758v2 (2008).

[20] D. Fuchs and S. Tabachnikov, Invariants of Legendrian and transverse knots in thestandard contact space, Topology Vol. 36 No. 5 (1997), 1025–1053.

[21] H. Geiges, An Introduction to Contact Topology, Cambridge studies in advanced math-ematics no. 109, Cambridge Univ. Press (2008).

[22] E. Giroux, Convexite en topologie de contact, Comment. Math. Helv. Vol. 66 (1991),615–689.

[23] E. Giroux, Geometrie de contact: de la dimension trois vers les dimensions superieures,Proceedings of the International Congress of Mathematicians Vol. II (Beijing, 2002), 405–414, Higher Ed. Press.

[24] K. Honda, On the classification of tight contact structures I, Geom. Topol. Vol. 4 (2000),309–368.

[25] K. Honda, On the classification of tight contact structures II, J. Differential Geom. Vol.55 (2000), 83–143.

[26] M. Hedden, Some remarks on cabling, contact structures, and complex curves, Proc. of13th Gokova Geometry-Topology Conference (2008), 1–11.

[27] M. Hedden, Notions of positivity and the Ozsvath-Szabo concordance invariant, e-printat arXiv:math/0509499 (2005).

[28] Y. Kanda, The classification of tight contact structures on the 3-torus, Comm. Anal.Geom. Vol. 5 No. 3 (1997), 413–438.

[29] P. Kronheimer and T. Mrowka, Witten’s conjecture and Property P, Geom. Topol. Vol.8 (2004), 295–310.

BIBLIOGRAPHY 153

[30] D. LaFountain, Studying uniform thickness I: Legendrian simple iterated torus knots,e-print at arXiv:0905.2760 (2009).

[31] D. LaFountain, Studying uniform thickness II: Transversally non-simple iterated torusknots, pre-print available upon request (2009).

[32] D. LaFountain and W. Menasco, Climbing a Legendrian mountain range without Sta-bilization, e-print at arXiv:0801.3475v2 (2009).

[33] H. Matsuda, A calculus on links via closed braids, preprint, 2007.

[34] H. Matsuda and W. Menasco On rectangular diagrams, Legendrian knots and trans-verse knots, e-print at arXiv:0708.2406 (2007).

[35] W. Menasco, On iterated torus knots and transversal knots, Geom. Topol. Vol. 5 No.21 (2001), 651–682.

[36] W. Menasco, Erratum: On iterated torus knots and transversal knots, Geom. Topol.Vol. 5 No. 21 (2001), 651–682.

[37] W. Menasco, An addendum on iterated torus knots, e-print at arXiv:math/0610566(2006).

[38] J. Milnor, Singular points of complex surfaces, Princeton Univ. Press, Princeton (1968).

[39] L. Ng, P. Ozsvath, and D. Thurston, Transverse knots distinguished by knot Floerhomology, J. Symplectic Geom. Vol. 6 No. 4 (2008), 461–490.

[40] S.Y. Orevkov and V.V. Shevchishin, Markov Theorem for transversal links, J. KnotTheory Ramifications Vol. 12 No. 7 (2003), 905–913.

[41] E. Pavelescu, Braiding knots in contact 3-manifolds, e-print at arxiv:0902.3715 (2009).

[42] T. Shibuya, Genus of torus links and cable links, Kobe J. Math. Vol. 6 No. 1 (1989),37-42.

[43] W.P. Thurston and H.E. Winkelnkemper, On the existence of contact forms, Proc.Amer. Math. Soc. Vol. 52 (1975), 345–347.

[44] B. Tosun, On the Legendrian and transverse classification of cablings, preprint (2009).

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