Neil Wright- Invariants of Knots and Links: Zeros of the Jones Polynomial

8/3/2019 Neil Wright- Invariants of Knots and Links: Zeros of the Jones Polynomial

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Invariants of Knots and Links:

Zeros of the Jones Polynomial

Neil Wright

Supervisor: Prof. E. Corrigan

April 29, 2010


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Abstract

Following an introduction to knot theory, we will look at the relationshipbetween the Jones polynomial and statistical mechanics. This serves as mo-tivation to study the zeros of the Jones polynomial for families of links. Wewill then use graph polynomials to find general expressions of the Jonespolynomial for some families of links. This will allow us to find the accumu-lation sets of the Jones polynomial zeros as the number of crossings in thelinks tends to infinity.


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Contents

1 Introduction to Knots and Links 2

2 The Jones Polynomial 9

3 The Tutte and Chain Polynomials of Graphs 16

4 Torus Links 22

5 Pretzel Links 24

6 Montesinos Links 28

7 Zeros of the Jones Polynomial 34

8 Conclusion 41

Bibliography 42

A A Knot Table 44

B Jones Polynomial Results 45

C Chain Polynomial Results 48

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Chapter 1

Introduction to Knots and

Links

Figure 1.1: A knot Figure 1.2: A link

Knot theory is the study of the ways in which we can tie a piece string, orhow a 1 dimensional string can lie in ordinary three-dimensional space[2]. In the late 19th century, a theory of Lord Kelvin was that atoms wereknotted, and so the properties of elements were related to the way in whichthe atoms were knotted [1]. This led in 1877 to P.G. Tait beginning theenumeration of knots: obtaining a table of knots such that knots whichwere equivalent (i.e. could be deformed into each other) only appear in thetable once [1]. See Appendix A for an example of a knot table.

The theory of invariants of knots and links is to find ways to establishwhether two given knots are equivalent, or are in fact distinct knots. Theprimary aim of research is to find better invariants that are able to distin-guish more knots, but there are other interesting aspects to knot invariantssuch as their relations to mathematics in other areas. In this report, we willlater concentrate on the relation between one particular invariant, the Jonespolynomial, and the Potts model used in statistical mechnics.

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1.1 Definition

In the definition of knots (and links) we want to capture the idea of aknotted loop (or in the case of links, loops) of rope. Essentially, a knot is anembedding of a circle, S2, into R3, but we wish to avoid some complications.Firstly, we want to avoid self-intersections in knots and links. Secondly wewant to avoid wild knots and links, where a knot or link contains an infinitenumber of kinks getting progressively smaller, as shown in Figure 1.3.

Figure 1.3: A wild knot

By defining knots and links via piecewise linear closed curves we can avoidthese complications:

Definition 1.1 [2]. A link ofm components is a subset ofS3, or ofR3, thatconsists of m disjoint, piecewise linear, simple closed curves. A link of onecomponent is a knot.

Although piecewise linear implies that the curves are made up of straightsegments, we generally assume there are so many of these segments thatimages and diagrams of knots and links are drawn with smoothly curvedcomponents [2]. Since the definition of links includes knots as a subset, wewill generally use the word link when referring to both (and only use knotwhen we want to refer specifically to one component links).

The unknot is the simplest knot, and is just the unkotted circle [3] (seeFigure 1.4).

Figure 1.4: The unknot

We can give links the addition structure of orientation. An oriented link isa link where each component is given a direction (i.e. each component isthe map of an oriented circle [4]).

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One property of a link diagram that will be used in later definitions is that

of writhe :

Definition 1.3 [2]. The writhe w(D) of a diagram D of an oriented link isthe sum of the signs of the crossings of D, where each crossing has sign +1or 1 as defined (by convention) in Figure 1.8.

+1 -1

Figure 1.8: The sign of a crossing in a link diagram

1.3.1 Reidemester moves

If we have two different diagrams of the same link (i.e. equivalent links),then the diagrams are related by the three Reidemeister moves and anorientation-preserving homeomorphism of the plane [2]. The orientation-preserving homeomorphism of the plane amounts to warping or changingthe link diagram (or parts of the link diagram) without altering the cross-ings. The three types of Reidemeister move are show in Figures 1.9, 1.10

and 1.11.

Figure 1.9: Type I Reidemeistermove

Figure 1.10: Type II Reidemeistermove

Figure 1.11: Type III Reidemeister move

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1.4 Prime knots

We can define the composition of knots as follows: Remove a small arc fromeach diagram of two knots and then join the loose ends [3] (see Figure1.12). The composition of two knots K and J may be denoted by K#J (asin [3]) or by K+ J (as in [2]). The composition of any knot with the unknotwill just give that knot [3].

KJ K # J

Figure 1.12: Composition of knots

A composite knot is a knot that can be formed by the composition of twoother knots (which are not the unknot). A knot that cannot be made in thisway is called a prime knot [3]. Appendix A shows the prime knots with upto 7 crossings.

If we have two oriented knots K and J then there are two ways to composethem: with their orientations corresponding (see Figure 1.13a); or with their

orientations not corresponding (see Figure 1.13b). In the first case, the knotK#J will be the same wherever we compose the knots. This is also true inthe seond case, but the knots formed in each case may not be the same [3].

K J K # J

(a) Corresponding orientation

K J K # J

(b) Opposite orientation

Figure 1.13: Composition of oriented knots

1.5 Invariants

The fundamental problem in knot theory is to distinguish links that are notequivalent, and to find when two (perhaps very different) diagrams actuallyrepresent the same link. A link invariant assigns to a link a mathmeticalobject (e.g. a number, a polynomial, a group) and gives the same result forequivalent links [2]. Equivalently, it will not change under the actions of thethree Reidemeister moves [1].

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1.5.1 Examples

The are many different link invariants, here we present some examples ofthe most commonly seen invariants.

Crossing number

The crossing number (or crossing index [1]) of a link is the minimum numberof crossings used in any diagram of the link [2]. The unkot has crossingnumber zero, while there is no knot with crossing number 1 (because any

way we join the four ends of a single crossing we obtain a diagram of a knotequivalent to the unkot) [3].

Unknotting number

The unknotting number of a link is the least number of crossings that needto be changed (i.e. the overpass and underpass swapped) in any diagram ofthe link to obtain a diagram of the unknot [1].

Colourability

A knot is colourable (or tricolourable) if we can assign one of three coloursto each arc in the knot diagram, such that [1]:

1. At least 2 colours are used.

2. At any crossing where 2 colours appear, all three colours appear.

(It does not matter which diagram of a knot we use, since if a knot iscolourable all diagrams of the knot will be colourable [1].) The unkot(see Figure 1.4) is not colourable, but the trefoil knot (see Figure 1.7) is

colourable [3].

Mod p Labelling

This is a generalisation of colourability, where the case p = 3 correspondsto the definition of colourability above [1]. A knot can be mod p labelled ifwe can assign to each arc in the knot diagram an integer from 0 to p 1,such that [1]:

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1. At least 2 labels are distinct.

2. At every crossing we have

2x y z = 0 (mod p)

where x is the label on the overpass, and y and z are the labels on thetwo parts of the underpass.

Genus

We can associate with a knot an orientable surface which has the knot as its

boundary. This surface is called a Seifert surface of the knot (it may not beunique) [1]. The genus of a knot is the minimum possible genus of a Seifertsurface of the knot.

Group

The group of a link is the fundamental group of the complement of the linkin R3 [4].

Polynomials

There are a number of link invariants that are polynomials (in one or morevariables). The first of these was introduced in 1928 by James Alexander,and is the Alexander polynomial [3]. It is a Laurent polynomial associatedwith oriented links [2]. In 1969 John Conway showed that Alexander poly-nomials could be calculated using only the polynomial of the unknot anda skein relation (a relation connecting the polynomials of three links whosediagrams differ at one crossing) [3]. He introduced the Conway polynomial,which can be obtained from the symmetric Alexander polynomial by a sub-

stitution of variables [1].

Many more polynomials based on such skein relations have been developed,including the Jones polynomial introduced in 1987 by Vaughan Jones in [5].

The Jones polynomial of an oriented link is a Laurent polynomial in t12 with

integer coefficients. In this report we are concerned with the Jones polyno-mial beacuse of its relationship to the mathematics of statistical mechanics.

The HOMFLY polynomial is a two-variable polynomial generalisation ofthe Alexander and Jones polynomials [3].

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Chapter 2

The Jones Polynomial

VL = t + t3 t4 VL = t

12 t

52

Figure 2.1: A knot and a link and their Jones polynomials

The Jones polynomial, V(L) or VL, of an oriented link L is a Laurent poly-

nomial in t12 with integer coefficients. It was introduced by V.F.R Jones in

1987 (see [5]). The Jones polynomial is calculated from the link diagramand is invariant under the Reidemester moves [2]. Two simple examples ofJones polynomials are given in Figure 2.1.

Although the Jones polynomial was first introduced via von Neumann al-gebras, it can be defined via the bracket polynomial of L.H. Kauffman (see[7]) as follows:

Defnition 2.1 [2]. The Jones polynomialV(L) of an oriented link L is the

Laurent polynomial in t 12 , with integer coefficients, defined by

V(L) =

(A)3w(D)Dt12=A2

Z

t12 , t

12

where D is any oriented diagram for L, w(D) is the writhe of D, and D isthe bracket polynomial of the unoriented diagram.

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2.1 The Kauffman Bracket Polynomial

The Kauffman bracket, D, assigns to an unoriented link diagram D aLaurent polynomial in A with integer coefficients [2]. It can be defined bythe three relations

(i) = 1,

(ii) D =

A2 A2

D,

(iii) = A + A1 ,

where the brackets in (iii) contain link diagrams that are identical except in

one place where they differ as shown [2].

Consider how the three Reidemeister moves affect the Kaufmann bracket:

Lemma 2.2 [2]. If a diagram is changed by a Type I Reidemeister move,its bracket polynomial changes in the following way:

= A3

= A3

Lemma 2.3 [2]. If a diagram D is changed by a Type II or Type III Rei-

demeister move, then D does not change.

The writhe of a link diagram will also not be changed by Type II or Type IIIReidemeister moves but a Type I Reidemeister move will change the writheby +1 or 1 [2]. In fact (for either possible orientation),

w ( ) = w ( ) + 1

w ( ) = w ( ) 1 .

Combining this with Lemmas 2.2 and 2.3 and Definition 2.1, we see that theJones polynomial is invariant under all three Reidemeister moves and so is

a link invariant [2].

2.1.1 An Example

An example calculation of a Jones polynomial via the Kauffman bracket:

= A + A1

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By Lemma 2.2 we have

= (A3)(A3) = A6 .

By Lemma 2.2 we also have

= A3 = A3 and

= A3 = A3 .

So we have

=A + A

1

=A(A3) + A1(A3)

= A4 A4 .

So

=A(A4 A4) + A1(A6)

= A5 A3 + A7 .

From the oriented diagram we have

w = 3So

(A)3w = A9(A5 A3 + A7)

=A4 + A12 + A16

V

=t + t3 t4 .

2.2 Properties

The Jones polynomial obeys the following skein relation [2]:t12 t

12

V

= t1V

tV

Where the link diagrams are identical except in one place where they differas shown.

If L is an oriented link with Jones polynomial V(L), substituting t1 for tgives the Jones polynomial for the mirror image of L [2].

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If K is an oriented knot formed by the composition of oriented knots K1

and K2, then [2]V(K) = V(K1) V(K2) .

2.3 Statistical Mechanics

In introducing the Jones polynomial [5], V.F.R. Jones established a rela-tionship between knot theory and statistical mechanics that has led to lotsof research [3]. The von Neumann algebra used in [5] was seen to be resem-ble the algebra of the Temperley-Lieb formulation of the Potts model [16].

The Yang-Baxter equation is another connection between knot theory andphysics, and is now used in knot theory although originating in physics [4].The parallel between the Star-Triangle relation and Type III Reidemeistermoves is another area of interest in the connection between knot theory andstatistical mechanics [16].

In this report we are interested in the direct relation between the Jonespolynomial and the partition function of the Potts model in statistical me-chanics (the Jones polynomial is given by a special case of the partitionfunction [18]). This motivates us to find general expressions of Jones poly-nomials for families of links, so that we may find the distribution of the zeros

of Jones polynomials as the number of crossings tends to infity.

2.3.1 The Potts Model

In statistical mechnics, the Ising model is used to study systems of particleswhere only particles close together interact [3] and we wish to study theoverall properties of the system [8]. The Ising model considers particlessituated at vertices on a graph, where each particle is in one of two spinsstates and interactions occur along the egdes of the graph (the graphs areoften lattices) [3]. The Potts model generalises the Ising model to allow each

particle be in any one of q states [3].

As in [8], let us consider a simple Potts model where the interaction energybetween particles is constant and only interactions between particles directlyconnected by an edge are considered. Given a graph G and a set S of qelements (the spin states), then assigning an element of S to each vertexof G we obtain a particular state of G [8]. Two possible Hamiltonians (theenergy of a particular state of G) are [8]

h1() = Ji,j

(i, j) and h2() = Ji,j

(1 (i, j))

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where is a state of G, i and j are vertices of G in spin states i, j, is

the Kronecker delta function and J is the interaction energy (if J > 0 themodel is feromagnetic and ifJ < 0 it is antiferromagnetic [8]). The partitionfunction of the model (from which we can study the thermodynamics of thesystem) is then [8]

Zi(G) =

e(hi())

where the sum is taken over all possible . Choosing i = 1 or 2 will give therespective Hamiltonians. And = 1kT, where T is the temperature of thesystem and k is the Boltzmann constant [8].

The Jones polynomial can be directly related to the partition function of

the Potts model; an exact relation between the two is given in [14] anddescribed again here. Firstly, consider the Potts model where we take theHamiltonian as h1(w) above, and set K = J. Now let K vary so thateach pair of interacting spins, i and j, has an interaction Ki,j. We get thefollowing partition function:

Z(G) =

exp

i,j

Ki,j (i, j)

=

i,j

eKi,j(i,j)

=

i,j

1 + eKi,j 1 (i, j)where the product is taken over all edges of G. This is exactly the Pottsmodel partition function given in [14]. The partition function is then amultivariate polynomial in q and eKi,j .

Now consider an oriented link with Jones Polynomial V(t) and its link di-agram. Shade alternate regions of the link diagram (there are two possibleshadings, the inverse of each other). Place a particle (i.e. a vertex of thegraph) in each shaded region and place interactions (i.e. edges of the graph)

along crossings. An interaction is K+ when the crossing is + and K whenthe crossing is , as shown in Figure 1.8. A simple example of this processis given in Figure 2.3. The Jones polynomial of the link is then related tothe partition function of the Potts model on the graph, Z(q; eK+ , eK), bythe relation [14]

V(t) = q(M+1)/2(t)Nt(3n++m+)/2Z

q; eK+ , eK

where n is the number of crossings in the shaded link diagram (as shownin Figure 2.2), m is the number of Kinteractions, N is the total numberof crossings in the link diagram (so N = n+ + n = m+ + m), M is the

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number of vertices of the graph (i.e. shaded regions of the link diagram),

q = t + 2 + t1, eK+ = (t)1 and eK

= t.

+

-

Figure 2.2: Sign given to crossings of the shaded link diagram

+

+

+

+

++

++

Figure 2.3: A shaded link diagram and the associated Potts model graph

If we choose the alternate shading on the link diagram, then we will in-terchange each K+ and K (so m+ and m will also interchange). Therelation will then obtain the same Jones polynomial but with an extra sign(1)N [14].

2.3.2 Zeros

In [12] and [13], Yang and Lee highlighted the study of zeros of partitionfunctions, in paticular to study phase transitions.

In [12] a monatomic lattice gas in a box of volume V (allowed to exchangeatoms with an external reservior, with a fixed temperature), and the grandpartition function are considered. The thermodynamic functions of pressureand density can be calculated from the partition function, and are given bythe limits of their average values as volume tends to infinity [12]. So thelocation of the zeros of the grand partition function as V will dictatewhere the thermodynamic functions remain analytic:

Their [the zeros of the grand partition function] distribution in

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the limit V gives the complete analytic behaviour of the

thermodynamic functions... [12]

Phase transitions will only occur where the density function is not analytic,so only at points where the zeros of the grand partition function convergeas V [12]. (In particular, if we consider the zeros in the complex planethen the axis corresponding to real, positive values of a physical variable areof interest. [12])

In [13] it is shown that the study of the Ising model in a magnetic field isequivalent to the lattice gas model, so the study of their thermodynamicproperties is equivalent. In particular, the partition function (in the Ising

model) is proportional to the grand partition function (in the lattice gasmodel) and the number of spin states corresponds to the volume of gas [13].

Given the importance of zeros of partition functions to thermodynamic prop-erties and the relation between the Jones polynomial and the Potts modelpartition function (given in Section 2.3.1), we are motivated to study thezeros of Jones polynomials. In particular, we wish to find accumulation setsof zeros as the number of crossings in a family of links tends to infinity. Thisappears to have been intially done by Wu and Wang in [14] and continuedby Chang and Shrock in [15] and then Jin and Zhang in [16], [17] and [18].

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Chapter 3

The Tutte and Chain

Polynomials of Graphs

To study the zeros of Jones polynomials of links, we first want to find generalexpressions for the Jones polynomials of families of links. An approach usedin [15] is to use the connection between the Jones polynomial of an alter-nating link and the Tutte polynomial of a graph. A connection between theTutte polynomial and the chain polynomial can be used to find Tutte poly-

nomials for famlies of related graphs, as used in [16]. This is the approachwe will use, but first we must establish some properties of the polynomialsand the relations connecting them.

3.1 The Associated Graph of a Link Diagram

The following method of assigning a planar graph to a connected, alternatinglink diagram is used in [15] and [16]. (Here we shall continue with thenotation of [16].) A link diagram will divide the plane into a number of

regions. At each crossing of a link diagram, we can assign each of the foursmall regions as an A-channel or B-channel as shown in Figure 3.1. We can

A

BA

A AB

B

B

Figure 3.1: Assignment of channels to a link diagram

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then call each region of the link diagram an A-(B-)region if all the channels

it contains are A-(B-)channels. Assign to each A-region a vertex and jointwo vertices if they lie either side of a crossing in the link diagram. We shallcall the planar graph that has been created the associated graph of the linkdiagram.

In this report we generally use standard diagrams for families of knots, sowe will obtain standard graphs for families of knots. We will denote theassociated graph obtained from the standard diagram of a link L by GL.

3.2 The Tutte Polynomial

The Tutte polynomial, T[G](x, y), is a two variable polynomial that weassociate with a graph G. We can define the Tutte polynomial via thefollowing rules of deletion and contraction of edges of a graph [8]:

1. If e is an edge of a graph G, and it is neither an isthmus (an edgewhich if deleted will increase the number of components of G) or aloop (an edge with the same vertex at its endpoints), then

T[G] = T[G e] + T[G/e] .

2. IfG consists of i isthmuses and j loops, then

T[G] = xiyj .

Where G e is the graph obtained from G by removing the edge e, and G/eis the graph obtained from G by contracting (i.e. removing and identify theend vertices of) the edge e. So we can calculate the Tutte polynomial ofa graph by removing and contracting edges until we have a graph of onlyisthmuses and loops (the order in which we do these steps does not matter[8]). However, later in the report we will be calculating Tutte polynomials forvery large graphs, so we will instead use the Tutte polynomials relationshipto another polynomial called the chain polynomial (see the next section).

We are interested in the Tutte polynomial because of its relation to theJones polynomial for alternating links. The following theorem connectingthe Tutte polynomial to the Jones polynomial is given in [16] and [15]:

Theorem 3.1. Suppose a link L admits a connected, alternating orientedlink diagram D with a A-regions, b B-regions and writhew. Then the Jones

polynomial of L is given by the Tutte polynomial of the associated graph Gof the diagram D:

VL(t) = (1)wt

ba+3w4 T[G]

t, t1

.

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3.3 The Chain Polynomial

3.3.1 Chain Graphs

As described in [16], we can obtain a chain graph from a graph (and thenuse the chain polynomial of a chain graph to calculate the Tutte polynomialof a graph). To obtain a chain graph from a graph:

1. Repeatedly remove vertices of degree two and join the two adjacentvertices with a new edge.

2. Assign to each edge a label (e.g. a, b) with an associated positive

integer (e.g. na = 2, nb = 5) that is equal to the number of edges thenew edge has replaced.

As noted above we will be obtaining standard graphs, GL, from the standarddiagram of a link L. We will denote by ML the chain graph obtained fromGL.

3.3.2 The Chain Polynomial

The chain polynomial, Ch[M](, {all labels}), is a polynomial that we as-

sociate with a chain graph M. It can be defined using flow polynomials ofsubgraphs, or chromatic polynomials of graphs obtained by inserting chainsof edges [21]. (It was actually introduced to study chromatic polynomials ofsuch graphs [18].) Chain polynomials can be calculated using the followingrelations, given in [18] and [21]:

1. IfG is a graph with no edges, then

Ch[G] = 1 .

2. If an edge labelled a is a loop (i.e. its endpoints are the same vertex)

of G, then Ch[G] = (a )Ch[G a] .

If an edge labelled a is not a loop of G, then

Ch[G] = (a 1)Ch[G a] + Ch[G/a] .

Where G a is the graph obtained from G by removing the edge labelleda, and G/a is the graph obtained from G by contracting (i.e. removing andidentify the end vertices of) the edge labelled a.

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We are interested in using chain polynomials to calculate Tutte polynomials

for families of graphs. The following theorem connecting the chain polyno-mial to the Tutte polynomial is given with proof in [16] (we use = 1 q):

Theorem 3.2. In Ch[M], if we replaceq by(1 x)(1 y), and replace eachlabela by xna for every chain a, a polynomial Ch[M] is obtained. Then

T[G](x, y) =1

(x 1)mn+kCh[M]

where m, n and k are the number of edges, vertices and components of thechain graph M, respectively.

3.3.3 Properties

Lemma 3.3 [19]. If a is an edge of (a chain graph) M let H be the graphobtained from M by deleting the edge a, and let K be the graph obtainedfrom M by contracting the edge a. Then

(i) Ch[H] is the coefficient of a in Ch[M].

(ii) Ch[K] is obtained from Ch[M] by putting a = 1.

We can relate the Chain polynomial of a 2-vetex-connected chain graph to

the Chain polynomials of two chain graphs from which it is built. Thefolowing lemma is given with proof (it is presented as a corollary of a moregeneral theorem) in [20].

Lemma 3.4. IfG is a 2-vertex-connected graph as shown in Figure 3.2, andG1 and G2 are formed from it as shown, then

Ch[G] = P1P2 1

A1A2

whereCh[G1] = P1z + A1 and Ch[G2] = P2z + A2 .

We can generalise this to inflating edges by a graph. By this we mean toremove an edge from one graph, and replace it by another graph with asimilarly labelled edge removed. For example in Figure 3.2, we would beinflating the edge z in G1 by G2, obtaining G.

Lemma 3.5. Let G1 be a chain graph with n edges labelled z and let G2be a chain graph with one edge labelled z, i.e.

Ch[G1] =ni=0

cizi and Ch[G2] = P z + A .

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z z

G

G1 G2

Figure 3.2: A graph G formed from two other graphs

If G is formed by inflating all edges labelled z in G1 by G2 then

Ch[G] =ni=0

ci

A

niPi .

Proof. See Appendix C, Section C.1.

Lemma 3.6. If G is formed by inflating all edges in a sheaf chain graph bya chain graph G2 with Ch[G2] = P z + A, then

Ch[G] =1

1

P +

A

n (P + A)n

.

Proof. Substituting the coefficients of z given in equation 3.3 into the equa-tion from Lemma 3.5 obtains

Ch[G] =ni=0

(1)n1

n

n i

ni

1

A

niPi

=1

1

P +

A

n (P + A)n

.

3.3.4 Useful Results

The following results for chain polynomials will be useful when we latercalculate Jones polynomials of families of links.

If M is a sheaf chain graph (i.e. a chain graph consisting of two vertices joined by n parallel edges) with edges labelled a1, a2, . . . , an, then [18]

Ch[M] =1

1

ni=1

(ai ) ni=1

(ai 1)

. (3.1)

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So,

Ch

ab

= 11

(b ) (a )n1 (b 1) (a 1)n1

(3.2)

where there are n 1 edges labelled a. (This result is given in [16].)

And,

Ch

z

=

1

1 [(z )n (z 1)n]

=ni=0

zi (1)n1

n

n i

ni

1

(3.3)

where there are n edges labelled z.

And,

Ch

ab

=1

1 [(b )n (a )n (b 1)n (a 1)n] (3.4)

where there are n edges labelled a and n edges labelled b.

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Chapter 4

Torus Links

In this chapter, we will study a family of links called the torus links. Af-ter establishing the definition and our notation, we will present the Jonespolynomials for torus knots and two types of torus links.

4.1 Definition and Notation

Torus links are knots and links that lie on the unkotted torus in S3 [3].We can define a torus link by the number of times it wraps around a torusin each direction: if it wraps around the meridian p times and around thelongitude q times then we call it the (p,q) torus link [3]. See Figure 4.1 fora standard diagram of the (p,q) torus link. We will denote the (p,q) toruslink by T(p,q).

q

p

{

{

Figure 4.1: A standard diagram of the (p,q) torus link

Some properties of torus links:

1. T(p,q) is equivalent to T(q, p) [3].

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2. If p and q are relatively prime (i.e. gcd(p,q) = 1) then T(p,q) is a

knot [3].3. Ifp = 1 or q = 1 (the other arbitrary) then T(p,q) is the unkot [4].

4.2 Jones polynomial

4.2.1 Torus Knots

The (p,q) torus link is a knot (i.e. it has one component) when p and q are

relatively prime. In this case the Jones polynomial is [2]

VT(p,q) =t(p1)(q1)

2

1 t2

1 tp+1 tq+1 + tp+q

. (4.1)

4.2.2 (p, q) Torus Links with p = 2, 3

Where p and q are not relatively prime, the (p,q) torus knot is a link of 2or more components.

If p = 2 then, for any q 1

VT(2,q) = (1)q+1

tq12 +

tq+12

1 + t((t)q + t)

. (4.2)

The proof of this is given in Appendix B, Section B.2.

The Jones polynomial for a (p,q) torus knot/link with p = 3 is given in [10]as:

VT(3,q) =tq1 + tq+1 + 2t2q if q 0 (mod 3),

tq1 + tq+1 t2q otherwise.(4.3)

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Chapter 5

Pretzel Links

In this chapter, we will study a family of links called the pretzel links.After establishing the definition and our notation, we will obtain generalexpressions for the Jones polynomials of three particular families of pretzellinks.


{p1 p2 pn{ {

Figure 5.1: General pretzel link Figure 5.2: (2,3,2,2,5) pretzel link

A pretzel link is formed by joining tangles in a cyclic fashion, where eachtangle consists of two vertical strings twisted together. We describe a pretzellink by the n-tuple of the number of half-twists (i.e. crossings) in each tangle(the sign given by Figure 5.3). The standard diagram of the (p1, p2,...,pn)pretzel link is given in Figure 5.1 and an example given in Figure 5.2.

Continuing the notation of [16], we will denote the (k1, k2,...,kn) pretzel link

by P(k1, k2,...,kn). Also denote the (

n k1,...,k1) pretzel link by P (k1(n)).

Similarly, denote the (

n k1,...,k1,

m k2,...,k2) pretzel link by P(k1(n), k2(m)).

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+ve -ve

Figure 5.3: Sign of half-twists in a pretzel link


5.2.1 (k(n)) Pretzel Links

In [16] the Jones polynomial (and its zeros) of P(3(n)) are studied. Here,we will use the same methods to study the more general case of P(k(n)) fork > 0.

{k k k{ { a a a

Figure 5.4: Diagram of P(k(n)) with associated graph, GP(k(n)), andassociated chain graph, MP(k(n)) (where there are n edges labelled aand na = k).

The pretzel link containing n tangles all with k > 0 half-twists, P(k(n)),admits an associated chain graph that is a sheaf graph (i.e. a graph con-sisting of two vertices joined by n edges) with every edge having the samelabel (see Figure 5.4). Then by equation 3.3 and Theorem 3.2 we have theTutte polynomial for the associated graph of P(k(n)):

T[GP(k(n))] =1

(x 1)n(y 1)

(xk x + xy y)n + (xk 1)n(xy x y)

.

By inspection of the link diagram we can see that a = 2 + (k 1)n andb = n. Then by Theorem 3.1 we have

VP(k(n)) =(1)kn t

2nkn2+3w4

(t 1)n (t1 1)

(t)k + t + 1 + t1n

+

t + 1 + t1

(t)k 1n

.

(5.1)

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Where w depends on the signs of k and n, and the orientation given to the

components of the link.

5.2.2 (k(1), l(n 1)) Pretzel Links

In [16] the Jones polynomial (and its zeros) of P(k(1), 1(n 1)) andP (k(1), 2(n 1)) are studied. Here, we will use the same methods to studythe more general case of P(k(1), l(n 1)) for l > 0 and k > 0.

{k l l{ { b a a

Figure 5.5: Diagram of P(k(1), l(n 1)) with associated graph,GP(k(1),l(n1)), and associated chain graph, MP(k(1),l(n1)) (where thereare n 1 edges labelled a, nb = k and na = l).

The pretzel link containing n tangles one with k half-twists and the remain-

ing with l half-twists, P(k(1), l(n 1)), admits an associated chain graphthat is a sheaf graph with all but one edge having the same label (see Figure5.5). Then by equation 3.2 and Theorem 3.2 we have the Tutte polynomialfor the associated graph of P(k(1), l(n 1)):

T[GP(k(1),l(n1))] =1

(x 1)n(y 1)

(xl x + xy y)n1(xk x + xy y)

+(xl 1)n1(xk 1)(xy x y)

.

By inspection of the link diagram we can see that a = l + nk k n + 2and b = n. Then by Theorem 3.1 we have

VP(k(1),l(n1)) =(1)wt

2nlnk+k2+3w4

(t 1)n(1 t1)

(t)l + t + 1 + t1

n1 (t)k + t + 1 + t1

+

(t)l 1n1

(t)k 1

t + 1 + t1

.

(5.2)

Where w depends on the signs of l, k and n, and the orientation given tothe components of the link.

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5.2.3 (k(n), l(n)) Pretzel Links

{k l l{ { b ak{Figure 5.6: Diagram ofP(k(n), l(n)) with associated graph, GP(k(n),l(n)),and associated chain graph, MP(k(n),l(n)) (where there are n edges la-belled a, n edges labelled b, nb = k and na = l).

The pretzel link containing 2n tangles half with k half-twists and half withl half-twists, P(k(n), l(n)), admits an associated chain graph that is a sheafgraph with half of the edges having the same label (see Figure 5.6). Thenby equation 3.4 and Theorem 3.2 we have the Tutte polynomial for theassociated graph of P(k(n), l(n)):

T[GP(k(n),l(n))] =1

(x 1)n(y 1)

xl x + xy y

n xk x + xy y

n+xl 1n xk 1n (xy x y) .

By inspection of the link diagram we can see that a = ln + kn 2n +2 andb = 2n. Then by Theorem 3.1 we have

VP(k(n),l(n)) =(1)wt

4nlnkn2+3w4

(t 1)n(1 t1)

(t)l + t + 1 + t1n

(t)k + t + 1 + t1n

+

(t)l 1n

(t)k 1n

t + 1 + t1

.

(5.3)

Where w depends on the signs of l, k and n, and the orientation given tothe components of the link.

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Chapter 6

Montesinos Links

In this chapter, we will study some of the family of links called the Mon-tesinos links. After establishing the definition and our notation, we willobtain general expressions for the Jones polynomials of two particular fam-ilies of Montesinos links.


6.1.1 Rational Tangles

Figure 6.1: An example (rational)tangle

Figure 6.2: An example (rational)tangle

A tangle is a link with four free ends which touch a bounding sphere [3],two examples are show in Figures 6.1 and 6.2. If we imagine the four freeends of a tangle to be restricted to the sphere, then a tangle that we candeform into two separate strands by only moving the free ends on thesphere is called a rational tangle [22]. (The tangle in Figure 6.2 is a rationaltangle.) All rational tangles are equivalent to a basic tangle that can be

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formed in the following way [22]:

1. Start with a horizontal (or vertical) tangle

2. Add a vertical (or horizontal) tangle to the bottom (or right)

3. Add a horizontal (or vertical) tangle to the right (or bottom)

4. Repeat these steps a finite number of times

(Where a horizontal (or vertical) tangle is formed by starting with two par-allel horizontal (or vertical) strands and twisting the endpoints. Each hori-zontal (or vertical) tangle has an associated integer equal to the number ofhalf-twists (or crossings) it contains, where sign depends as shown in Figure

6.3.) When we start with a horizontal (or vertical) tangle, then the resultis called a basic horizontal (or vertical) tangle [22].

A horizontal tangle, with associated integer +4.

A horizontal tangle, with associated integer -3.

A vertical tangle,

with associated integer +3.

A vertical tangle,

with associated integer -2.

Figure 6.3: Horizontal and vertical tangles

In this report we will use two particular rational tangles, shown in Figures

6.4 and 6.5. We will denote the rational tangle shown in Figure 6.4 by {k, l}and in Figure 6.5 by {m,k,l}.

6.1.2 Montesinos Links

The Montesinos links are formed in a similar way to the pretzel links, buteach tangle consists of a rational tangle. The standard diagram for the(classic) Montesinos link is given in Figure 6.6, where each box (T1, . . . , T n)contains a rational tangle.

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k{

l{

Figure 6.4: {k, l} rational tangle

l{

k{m

{

Figure 6.5: {m,k,l} rational tan-gle

T1 T2 Tn

Figure 6.6: Standard diagram for a (classic) Montesinos link

In this report we will be using Montesinos links where all of the n tanglesare the same. We will call the Montesinos link where every tangle is the

{k, l} tangle (Figure 6.4), the ({k, l}(n)) Montesinos link and denote it by

M({k, l}(n)). Similarly, where every tangle is the {m,k,l} tangle (Figure6.5), it is the ({m,k,l}(n)) Montesinos link and denote it by M({m,k,l}(n)).


6.2.1 ({k, l}(n)) Montesinos Links

In the family of links given by M({k, l}(n)) each tangle is as shown inFigure 6.4. The associated graph of M({k, l}(n)) consists of n copies of

the subgraph shown in Figure 6.7, with all vertices labelled 1 identifiedand all vertices labelled 2 identified. Then the associated chain graph ofM({k, l}(n)) consists of n copies of the subgraph shown in Figure 6.8, withall vertices labelled 1 identified and all vertices labelled 2 identified.

To find a general expression for the Jones polynomial of these links, firstwe obtain the chain polynomial of a graph from which we can build the

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with A = (t)k+2 (t)k+1 + (t)k and B = (t)k+1. Where w depends

on the signs of l, k and n, and the orientation given to the components ofthe link.

6.2.2 ({m,k,l}(n)) Montesinos Links

In the family of links given by M({m,k,l}(n)) each tangle is as shown inFigure 6.5. The associated graph of M({m,k,l}(n)) consists of n copiesof the subgraph shown in Figure 6.9, with all vertices labelled 1 identifiedand all vertices labelled 2 identified. Then the associated chain graph ofM({m,k,l}(n)) consists of n copies of the subgraph shown in Figure 6.10,

with all vertices labelled 1 identified and all vertices labelled 2 identified.

1

2

Figure 6.9: Subgraph of GM({m,k,l}(n))

a

b1

2

c

Figure 6.10: Subgraph of

MM({m,k,l}(n)) (na = k, nb = 1,nc = 1, l edges labelled b, m edgeslabelled c)

To find a general expression for the Jones polynomial of these links, firstwe obtain the chain polynomial of a graph from which we can build theassociated chain graph of M({m,k,l}(n)):

Ch

a

z

b

c

=

z

(1 )2

a

(b )l (b 1)l

((c )m (c )m)

+(b )l + (b 1)l ((c )m (c 1)m)+

1

(1 )2

(b )l + (b 1)l

((c )m (c 1)m)

1

(1 )2

(1 + )((b )l + (b 1)l)

a((b )l (b 1)l)

((c )m (c 1)m)

Where there are l edges labelled b and m edges labelled c. (The calculation

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of this is given in Appendix C, Section C.2.) Then by Lemma 3.6 we obtain:

Ch[MM({m,k,l}(n))] =1

1

(b 1)l(a 1) + (b )l(1 a)

(c 1)m

1

n

1

1

(b 1)l(a ) + (b )l( a)

(c )m

1

n

Setting na = k, nb = 1 and nc = 1, by Theorem 3.2 we have the Tuttepolynomial for the associated graph of M({m,k,l}(n)):

T[GM({m,k,l}(n))] =

(1)(l+m+1)n

(1 x)(l+m+1)n(1 y)n+1

((x 1)l(xk(x + y xy) 1) (xy y)l(xk 1))(x 1)mn

(x + y xy)((x 1)l(xk 1)(x + y xy) (xy y)l(xk (x + y xy)))(xy y)m

nBy inspection of the link diagram we can see that a = kn+2 and b = (l+m)n.Then by Theorem 3.1 we have

VM({m,k,l}(n)) =

(1)

(l+m)n++1

t

(l+mk)n2+3w4

tmn(t + 1)(l+2)n+1

((t 1)l(A + t) (1 + t1)l(B + t))(t)mn

(t2 + t + 1)

+t

(t 1)l(A t2 t 1) (1 + t1)l(B t2 t 1)n

(6.2)

with A = (t)k+2 (t)k+1 + (t)k and B = (t)k+1. Where w dependson the signs of l, k and n, and the orientation given to the components ofthe link.

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Chapter 7

Zeros of the Jones

Polynomial

As discussed in Chapter 2, the link between the Jones polynomial and thePotts model partition function (and the importance of zeros of partitionfunctions to thermodynamic properties) motivates us to study zeros of Jonespolynomials. In particular we wish to find the accumulation sets of zeros forfamilies of links as the number of crossings tends to inifinity. In this chapter

we will use the general expressions for Jones polynomials found in Chapters4, 5 and 6 to find some of these accumulation sets.

Whilst the Jones polynomial is not a true polynomial it can always be ex-pressed as a polynomial in t multiplied by some factor [5], so we are justifiedin finding zeros [18]. We will only consider the non-zero zeros of Jones poly-nomials (so we can assume t = 0 in our calculations) and our plots willalso not include any t = 0 solutions to numerical examples. Also, sincethe Jones polyomial of a knot does not depend on orientation [2] and theJones polynomials of links only differer by a factor of ta (for some integer a)for different orientations [16], we do not need to consider orientation when

finding zeros in this chapter.

7.1 General Knots and Links

The Jones polynomial has the property [6]

VL(e2i3 ) = (1)n1

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Taking the qth root, gives us

t =

t + t121q

, = 1, 2,...,n,

with = e2in . Now as q we have t

n . So the zeros of a link

VT(3,q) are distributed on the unit circle as q . A numerical example(with q = 36) and the unit circle are shown in Figure 7.4.

Figure 7.1: Zeros ofVT(2,29) and ofVT(p,q)

as q for any fixed pFigure 7.2: Zeros ofVT(5,32) and ofVT(p,q)

as q for any fixed p

Figure 7.3: Zeros of VT(2,q) withq = 40 as q

Figure 7.4: Zeros of VT(3,q) withq = 36 and as q

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7.3 Pretzel Links

7.3.1 (k(n)) Pretzel Links

Equating the numerator of equation 5.1 to zero and rearranging (and weassume t = 0), we obtain

(t)k + t + 1 + t1

(t)k 1

n= (t + 1 + t1) .

Taking the nth root and n , we obtain the set

|(t)k + t + 1 + t1| = |(t)k 1|

on which the zeros of VP(k(n)) are distributed as n . An example of thecurves in the complex plane given by the set with k = 4 is given in Figure7.5 and with k = 5 in Figure 7.6.

Figure 7.5: Zeros of VP(4(n)) withn = 20 and as n

Figure 7.6: Zeros of VP(5(n)) withn = 20 and as n

7.3.2 (k(1), l(n 1)) Pretzel Links


(t)l + t + 1 + t1

(t)l 1

n1=

((t)k 1)(t + 1 + t1)

(t)k + t + 1 + t1.

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Taking the (n 1)th root and n , we obtain the set

|(t)l + t + 1 + t1| = |(t)l 1|

on which the zeros of VP(k(1),l(n1)) are distributed as n . This resultis identical to that for (l(n)) pretzel links, found in the previous section.

7.3.3 (k(n), l(n)) Pretzel Links


((t)l + t + 1 + t1)((t)k + t + 1 + t1)

((t)l 1)((t)k 1)

n= (t + 1 + t1) .

Taking the nth root and n , we obtain the set

|[(t)k + t + 1 + t1][(t)l + t + 1 + t1]| = |[(t)k 1)][(t)l 1]|

on which the zeros of VP(k(n),l(n)) are distributed as n . An examplewith k = 2 and l = 4 is given in Figure 7.7 and with k = 4 and l = 5 inFigure 7.8.

Figure 7.7: Zeros of VP(2(n),4(n))with n = 10 and as n

Figure 7.8: Zeros of VP(4(n),5(n))with n = 10 and as n

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7.4 Montesinos Links

7.4.1 ({k, l}(n)) Montesinos Links


(t 1)l(A + t) (1 + t1)(B + t)

(t 1)l(A t2 t 1) (1 + t1)(B t2 t 1)

n=

t

t2 + t + 1.

with A = (t)k+2 (t)k+1 + (t)k and B = (t)k+1. Taking the nth root

and n , we obtain the set

|(t1)l(A+t)(1+t1)(B+t)| = |(t1)l(At2t1)(1+t1)(Bt2t1)| .

Dividing by (1 + t1) we obtain the set

|(t)l[A + t] [B + t]| = |(t)l[A t2 t 1] [B t2 t 1]|

on which the zeros of M({k, l}(n)) are distributed as n . An examplewith l = 2 and k = 3 is given in Figure 7.9 and with l = 4 and k = 5 inFigure 7.10.

Figure 7.9: Zeros of VM({3,2}(n))with n = 15 and as n

Figure 7.10: Zeros of VM({5,4}(n))with n = 10 and as n

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Chapter 8

Conclusion

After introducting the basic ideas of knot theory, we looked at the relationbetween the Jones polynomial and the partition function of the Potts mdel.This motivated us to continue the investigation of zeros of the Jones Polyno-mial. In particular, we found general expressions for the Jones polynomialfor families of links, so that we could find accumulation sets of zeros of theJones polynomial as the number of crossings in the links tended to infinity.

The work of Wu and Wang [14], Chang and Shrock [15] and Jin and Zhang[16], [17] and [18], and this report concentrate on important families oflinks: the torus links, the pretzel links and the Monesinos links. Thereare still many links in these families whose Jones polynomials have notbeen studied in this way, and future investigation into these is possible. Inparticular, Montesinos knots containing more complex rational tangles andeven different rational tangles could be investigated.

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Appendix A

A Knot Table

Figure A.1: A knot table

Figure A.1 is reproduced from http://en.wikipedia.org/wiki/File:Knot_table.svg (Author: Jkasd, 2008).

Figure A.1 is a table of the prime knots (not including mirror images) withup to 7 crossings. The knots are labelled with the number of crossings, andthe subscript just denotes an order of the knots.

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Appendix B

Jones Polynomial Results

B.1 Jones polynomials of T(2,1) and T(2,2)

Figure B.1: T(2, 2) Figure B.2: T(2, 1)

By Lemma 2.2 we have:

= A3 = A3

= A3 = A3

Then using the definition of the Kauffman bracket:

=A + A1

=A A3 + A1 A3= A4 A4

The writhe of the oriented link is clear from the diagram:

w

= 2

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So by the definition of the Jones polynomial via the Kauffman bracket:

VT(2,2) = V

=

t14

32

t14

4

t14

4=t

32

t t1

= t12 t

52

(B.1)

By a Type I Reidemeister move on the diagram shown on Figure B.2 we caneasily see that T(2, 1) is equivalent to the unkot, so:

VT(2,1) = 1 (B.2)

B.2 Proof of equation 4.2

Proof. From equations B.1 and B.2 we can see that

VT(2,2) = t12 t

52 = (1)2+1

t212 +

t2+12

1 + t((t)2 + t)

and

VT(2,1) =1 = (1)1+1

t112 +

t1+12

1 + t((t)1 + t)

.

Given the skein relation of the Jones polynomial, we have

(t12 t

12 )VT(2,q1) = t

1VT(2,q) tVT(2,q2) . (B.3)

Assume we have q 1 and q 2 such that

VT(2,q1) =(1)(q1)+1

t(q1)1

2 +t(q1)+1

2

1 + t((t)(q1) + t)

and

VT(2,q2) =(1)(q2)+1t (q2)12 + t(q2)+1

2

1 + t ((t)(q2) + t) .

(B.4)

Then substituting B.4 into B.3 obtains

(t12 t

12 )(1)(q1)+1

t(q1)1

2 +t(q1)+1

2

1 + t((t)(q1) + t)

=t1VT(2,q) t(1)(q2)+1

t(q2)1

2 +t(q2)+1

2

1 + t((t)(q2) + t)

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VT(2,q) =t(1)

q tq12

+

tq+12

1 + t ((t)

q1

+ t) t(1)q

tq32 +

tq12

1 + t((t)q1 + t)

+ t(1)q1

tq12 +

tq+12

1 + t((t)q2 + t)

=(1)q

tq+12 +

tq+32

1 + t((t)q1 + t)

(1)q t q12 + tq+12

1 + t((t)q1 + t)

(1)q

tq+12 +

tq+32

1 + t((t)q2 + t)

=(1)q

tq12 +

1

1 + t

(t)q1t

q+32 + t

q+52

(t)q1tq+12 t

q+32 (t)q2t

q+32 t

q+52

=(1)q

t

q12 +

tq+32

1 + t (t)q1 1 (t)q2 + (t)q2

=(1)q

tq12

tq+12

1 + t((t)q + t)

=(1)q+1

tq12 +

tq+12

1 + t((t)q + t)

So, by induction,

VT(2,q) = (1)q+1

tq12 +

tq+12

1 + t((t)q + t)

for all q 1.

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Appendix C

Chain Polynomial Results

This appendix will include calculations on Chain polynomials that are notincluded in the main text.

C.1 Proof of Lemma 3.5

Proof. Let G(k) be the graph formed by inflating k edges labelled z in G1by G2, so G

(n) is G. We have

Ch[G1] =ni=0

cizi =

ni=1

cizi1

z + c0 .

So by Lemma 3.4

Ch[G(1)] =

ni=1

cizi1

P

A

c0

=ni=2

cizi2P z + P c1 A c0 .

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Then repeatedly applying Lemma 3.4 we obtain

Ch[G(2)] = n

i=2

cizi2

P2 A

P c1

A

c0

=

ni=3

cizi3

P2z + P2c2

A

P c1 +

A

2c0

Ch[G(3)] =

ni=3

cizi3

P3

A

P2c2

A

P c1 +

A

2c0

=

n

i=4ciz

i4

P3z + P3c3

A

P2c2 +

A

2P c1 +

A

3c0

...

Ch[G(n)] =Pncn +

A

Pn1cn1 +

A

2Pn2cn2

+ . . . +

A

n1P c1 +

A

nc0

=n

i=0ci

A

ni

Pi .

C.2 Calculation of Some Chain Polynomials

b

a

c

d

e

f

Figure C.1: K4 graph

a

c

d

Figure C.2: K3 graph

The chain polynomial of the complete graph with 4 vertices, K4 (labelled asshown in Figure C.1), is given in [19] as

Ch[K4] =abcdef (adf + abc + bef + cde) (ae + bd + cf)

+ ( + 2)(a + b + c + d + e + f) (2 + 32 + 3) .

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The complete graph with 3 vertices, K3 (labelled as shown in Figure C.2),

can be formed from K4 by removing edges b and e and contracting edge f.So by Lemma 3.3 we obtain

Ch[K3] = acd . (C.1)

By equation 3.2 we also have

Ch

bc

= 11

(c )(b )l (c 1)(b 1)l

=c

1 (b )l (b 1)l

+1

1

(b )l + (b 1)l

(C.2)

where there are l edges labelled b.

Applying Lemma 3.4 to equations C.1 and C.2 (and relabelling d as z) weobtain

Ch

a zb

=

za

1 (b )l (b 1)l

1

()

1

1

(b )l + (b 1)l

=

za

1

(b )l (b 1)l

+

1

1

(b )l + (b 1)l

(C.3)

where there are l edges labelled b.

By equation 3.1 we have

Ch

czz

= 1

1 [(c )m (z )2 (c 1)m (z 1)2] (C.4)

where there are m edges labelled c. Aplying Lemma 3.4 to equations C.3

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and C.4 we obtain

Ch

a

z

b

c

= z(1 )2

a

(b )l (b 1)l

((c )m (c )m)

+

(b )l + (b 1)l

((c )m (c 1)m)

+1

(1 )2

(b )l + (b 1)l

((c )m (c 1)m)

1

(1 )2

(1 + )((b )l + (b 1)l)

a((b )l

(b 1)l

) ((c )m (c 1)m)where there are l edges labelled b and m edges labelled c.

Documents

Neil Wright- Invariants of Knots and Links: Zeros of the Jones Polynomial