Kenneth L. Baker [email protected]/Zacatecas/zacatecas... · 2007. 5. 31. · Kenneth L. Baker (GaTech) Lens space ﬁllings of OPT–bundles AMS-SMM ’07 10 / 26. Problem Extend

Lens space fillings of once-punctured torus bundles

Kenneth L. [email protected]

Georgia Institute of TechnologyAtlanta, Georgia, USA

VII Reunión Conjunta AMS-SMM26 de mayo de 2007

Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 1 / 26

Lens space fillings of OPT–bundles: ∆ = 1 and ∆ > 1

TheoremM is an OPT–bundle admitting a lens space filling of distance ∆

m

M ∼= F × I/φ with φ = τnx τ2

y τmx τ−1

y for some m, n ∈ Z.

Moreover, ∆ = 1 with the following exceptions:

φ = τxτy and ∆ = | − 6k + 1| for k ∈ Z

φ = τ2x τy and ∆ = | − 4k + 1| for k ∈ Z

φ = τ3x τy and ∆ = | − 3k + 1| for k ∈ Z

φ = τ4x τy and ∆ = 1, 3

φ = τ5x τy and ∆ = 1, 2




m


y τmx τ−1



φ = τxτy and ∆ = | − 6k + 1| for k ∈ Z

φ = τ2x τy and ∆ = | − 4k + 1| for k ∈ Z

φ = τ3x τy and ∆ = | − 3k + 1| for k ∈ Z

φ = τ4x τy and ∆ = 1, 3

φ = τ5x τy and ∆ = 1, 2


Theorem (Burde & Zieschang, González-Acuña)

In S3 there are only two genus one fibered knots.(Up to homeomorphism)

The Trefoil The Figure Eight

(. . . but infinitely many genus one knots.)

















Definition

A knot K ⊂ M̂closed

is an OPT–knot if there is an essential

Once-Punctured Torus F

properly embedded in its exterior M = M̂ − N(K ).

DefinitionLet µ be the meridian of K . Then ∆ = ∆(µ, ∂F ) = |µ · ∂F | is

∆ = 8

the distance of µ and ∂F ,

the distance of the filling ofthe exterior M by N(K ), and

the order of K in H1(M̂).


Definition






∆ = 8





Definition






∆ = 8





Definition






∆ = 8





Definition






∆ = 8





Definition






∆ = 8





Definition






∆ = 8



the order of [K ] in H1(M̂).


∆ = 1

=⇒ K is null homologous

∆[K ] = [K ] = 0 ∈ H1(M̂; Z)

ZOPT–knot

∆ > 1

=⇒ K is rationally nullhomologous

∆[K ] = 0 ∈ H1(M̂; Z)

[K ] = 0 ∈ H1(M̂; Q)

QOPT–knot


∆ = 1

=⇒ K is null homologous

∆[K ] = [K ] = 0 ∈ H1(M̂; Z)

ZOPT–knot

∆ > 1

=⇒ K is rationally nullhomologous

∆[K ] = 0 ∈ H1(M̂; Z)

[K ] = 0 ∈ H1(M̂; Q)

QOPT–knot


DefinitionIf the exterior of K is a once-punctured torus bundle (OPT–bundle )

F × I/φ

then K is a OPT–fibered knot :ZOPT–fibered knot if ∆ = 1 (genus one fibered knot)QOPT–fibered knot if ∆ > 1



F × I/φ




F × I/φ




F × I/φ




F × I/φ




F × I/φ



The monodromy φ is acomposition of Dehn twistsalong basis curves x , y on F .

τx : 7→

τy : 7→



τx : 7→

τy : 7→



τx : 7→

τy : 7→


Example

The Trefoil

φ = τxτy

The Figure Eight

φ = τ−1x τy .

These are the two OPT–bundles that may be filled to produce S3.∆ = 1.


Example

The Trefoil

φ = τxτy

The Figure Eight

φ = τ−1x τy .

These are the two OPT–bundles that may be filled to produce S3.∆ = 1.


Definition

The Lens spaces L(p, q) aretwo solid tori glued togetheralong their boundaries.

To form L(p, q), the meridian ofone becomes a (p, q) curve onthe other.

Example

S3 ∼= L(1, n) for n ∈ Z RP2 ∼= L(2, 1) S1 × S2 ∼= L(0, 1)

L(p, q) ∼= L(p′, q′) ⇐⇒ |p′| = |p| and q′ ≡ ±q±1 mod p


ProblemExtend classification of OPT–fibered knots to lens spaces.

Morimoto began the classification of ZOPT–fibered knots in lensspaces up to homeomorphism:

L(m, 1) contains at least two if m > 0,exactly two if m ∈ {1, 2, 3, 5, 19},

L(4, 1) contains exactly three,L(0, 1), L(5, 2), and L(19, 3) contain exactly one,L(19, 2), L(19, 4), and L(19, 7) contain none.

=⇒ determined some OPT–bundles with lens space fillings.














QuestionWhat are the OPT–fibered knots in lens spaces?What OPT–bundles may be filled to produces a lens space?

ApproachFor ∆ = 1, we classify ZOPT–fibered knots in lens spaces.

(. . . infinitely many ZOPT–knots.)

For ∆ > 1, we classify QOPT–knots in lens spacesand then observe which are fibered.

(. . . infinitely many QOPT–knots?)
































∆ = 1

Idea

ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.

Example (The Figure Eight)


∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

Idea




∆ = 1

1–1 Correspondence

A closed 3–manifold M̂with ZOPT–fibered knot K

(M̂, K )↔

A link L in S3

with double branched cover M̂and an axis (unknot) A

that presents L as closed3–braid(L, A)

↔


∆ = 1



(M̂, K )↔

A link L in S3



↔


∆ = 1



(M̂, K )↔

A link L in S3



↔


∆ = 1

For lens spaces M̂ = L(p, q)

[Hodgson & Rubinstein] Lens spacesare only double branched covers of2–bridge links.

[Murasugi, Stoimenow] The 2–bridgelinks of braid index 3 or less are thoseshown.

[Birman & Menasco] The braid axes oflinks with braid index at most 3 areclassified.

When considered up tohomeomorphisms, we have the shown3–braid presentations of 2–bridge links.

All have the form σnx σ2

yσmx σ−1

y


∆ = 1







yσmx σ−1

y


∆ = 1







yσmx σ−1

y


∆ = 1







yσmx σ−1

y


∆ = 1







yσmx σ−1

y


∆ = 1







yσmx σ−1

y


∆ = 1

TheoremUp to homeomorphism, the lens space L(p, q) contains exactly

three ZOPT–fibered knots ⇐⇒ L(p, q) ∼= L(4, 1),

two ZOPT–fibered knots ⇐⇒ L(p, q) ∼= L(r , 1) for r > 0, 6= 4,

one ZOPT–fibered knot ⇐⇒ L(p, q) ∼= L(r , s) for either r = 0or for 0 < s < r where either

r = 2mn + m + n and s = 2n + 1 for some integers n, m > 1, orr = 2mn +m +n +1 and s = 2n +1 for some integers n, m > 0, and

no ZOPT–fibered knots otherwise.

TheoremThe OPT–bundle exterior of any ZOPT–fibered knot in a lens spacehas monodromy of the form

φ = τnx τ2

y τmx τ−1

y .


∆ = 1

TheoremUp to homeomorphism, the lens space L(p, q) contains exactly

three ZOPT–fibered knots ⇐⇒ L(p, q) ∼= L(4, 1),

two ZOPT–fibered knots ⇐⇒ L(p, q) ∼= L(r , 1) for r > 0, 6= 4,

one ZOPT–fibered knot ⇐⇒ L(p, q) ∼= L(r , s) for either r = 0or for 0 < s < r where either

r = 2mn + m + n and s = 2n + 1 for some integers n, m > 1, orr = 2mn +m +n +1 and s = 2n +1 for some integers n, m > 0, and

no ZOPT–fibered knots otherwise.

TheoremThe OPT–bundle exterior of any ZOPT–fibered knot in a lens spacehas monodromy of the form

φ = τnx τ2

y τmx τ−1

y .


∆ > 1

For ∆ > 1, the preceding argument does not work so well.

Instead we examine how a QOPT–knot K with OPTF interacts with aHeegaard torus T̂ of the lens space L(p, q).

If K is a torus knot (scc on T̂ ), then straightforward.Assume K is not a torus knot, then:

Put K into thin position with respect to T̂ .Show that K must be 1–bridge with respect to T̂ .Show that ∆ = 2 or ∆ = 3.Analyze resulting possible configurations.


∆ > 1






∆ > 1






∆ > 1






∆ > 1






∆ > 1

DefinitionK ⊂ L(p, q) is in thin position if it minimizes this count:

Given a height function h : L(p, q)→ [−∞,+∞]

so that h−1(•) is a

{torus if • ∈ (−∞,+∞)

circle if • = ±∞for a • in each interval of regular values of h(K ),

add up |h−1(•) ∩ K |.

Thin position is a “tightened” bridge position.

With K in thin position, take Heegaard torus T̂ = h−1(0)where 0 is just above a minimum of K and just below a maximum of K .

t = |T̂ ∩ K |

T = T̂ −N(K )← delusions of being essential.


∆ > 1




{torus if • ∈ (−∞,+∞)


add up |h−1(•) ∩ K |.



t = |T̂ ∩ K |



∆ > 1




{torus if • ∈ (−∞,+∞)


add up |h−1(•) ∩ K |.



t = |T̂ ∩ K |



∆ > 1


2

4

2

4

2

14



{torus if • ∈ (−∞,+∞)


add up |h−1(•) ∩ K |.



t = |T̂ ∩ K |



∆ > 1


2

4

2

4

2

14



{torus if • ∈ (−∞,+∞)


add up |h−1(•) ∩ K |.



t = |T̂ ∩ K |



∆ > 1




{torus if • ∈ (−∞,+∞)


add up |h−1(•) ∩ K |.



t = |T̂ ∩ K |



∆ > 1




{torus if • ∈ (−∞,+∞)


add up |h−1(•) ∩ K |.



t = |T̂ ∩ K |



∆ > 1

F = OPTand T = T̂ − N(K ) intersect nicely:

Form graphs GF on F and GT on T where:

Vertices are the punctures, andEdges are the arcs of F ∩ T .


∆ > 1





∆ > 1





∆ > 1

Thin position of K with choice of T̂=⇒ no monogons!

Vertices of GT have valence ∆=⇒ there are ∆t/2 Edges (on each GT and GF ).

Numbering the Vertices of GT in the order that K punctures T̂=⇒ numbering the endpoints of Edges of GF .


∆ > 1





∆ > 1





∆ > 1

A priori, the graph GF looks like one of the following:


∆ > 1

When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.

These “bind” too much of K=⇒ K is not in thin position.

With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)


∆ > 1





∆ > 1





∆ > 1





∆ > 1





∆ > 1





∆ > 1





∆ > 1





∆ > 1





∆ > 1





∆ > 1





∆ > 1

With t = 2, each face of GF is one of the following:


∆ > 1



∆ > 1



∆ > 1



∆ > 1

Studying how these pieces may fit together, we find two possibilities

∆ = 2GF GT

∆ = 3GF GT

In both situations, F is disjoint from a torus knot J

=⇒ meridians are not completely determined.


∆ > 1


∆ = 2GF GT

∆ = 3GF GT




∆ > 1


∆ = 2GF GT

∆ = 3GF GT




∆ > 1


∆ = 2GF GT

∆ = 3GF GT




∆ > 1

Get families of lens spaces with OPT–knots for ∆ = 2 and ∆ = 3.These torus knots J are OPT–knots, too.Exterior of K ∪ J is the exterior of the Whitehead link, W (·, ·).

TheoremUp to homeomorphism,K is a QOPT–knot in a lens space ⇐⇒ for k 6= 0 it is the core of oneof the following surgeries (except the −1 surgery)

The OPT–torus knots are the cores of the 6 + 1/k , −4 + 1/k , and−3 + 1/k surgeries.


∆ > 1





∆ > 1





∆ > 1





∆ > 1



−2

−4+1/k

−3

−3+1/k



∆ > 1



−1

−6+1/k

−2

−4+1/k

−3

−3+1/k



∆ > 1

FactW (γ, ·) is a OPT–bundle ⇐⇒ γ ∈ Z

TheoremIf K is a QOPT–fibered knot with monodromy φ in a lens space,then for k 6= 0 we have

K ⊂

L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)

L(12, 5)L(10, 3)

and φ ∼=

τxτy

τ2x τy

τ3x τy

τ4x τy

τ5x τy

.

← Trefoil← torus knot← torus knot← Klein bottle← Fig. Eight Sis.


∆ > 1


−1

−6+1/k

−2

−4+1/k

−3

−3+1/k


K ⊂

L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)

L(12, 5)L(10, 3)

and φ ∼=

τxτy

τ2x τy

τ3x τy

τ4x τy

τ5x τy

.



∆ > 1


−1

−6+1/k

−2

−4+1/k

−3

−3+1/k


K ⊂

L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)

L(12, 5)L(10, 3)

and φ ∼=

τxτy

τ2x τy

τ3x τy

τ4x τy

τ5x τy

.



∆ > 1


−1

−6+1/k

−2

−4+1/k

−3

−3+1/k


K ⊂

L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)

L(12, 5)L(10, 3)

and φ ∼=

τxτy

τ2x τy

τ3x τy

τ4x τy

τ5x τy

.



∆ > 1


−1

−6+1/k

−2

−4+1/k

−3

−3+1/k


K ⊂

L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)

L(12, 5)L(10, 3)

and φ ∼=

τxτy

τ2x τy

τ3x τy

τ4x τy

τ5x τy

.



∆ > 1


−1

−6+1/k

−2

−4+1/k

−3

−3+1/k


K ⊂

L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)

L(12, 5)L(10, 3)

and φ ∼=

τxτy

τ2x τy

τ3x τy

τ4x τy

τ5x τy

.



∆ > 1


−1

−6+1/k

−2

−4+1/k

−3

−3+1/k


K ⊂

L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)

L(12, 5)L(10, 3)

and φ ∼=

τxτy

τ2x τy

τ3x τy

τ4x τy

τ5x τy

.



∆ > 1


−1

−6+1/k

−2

−4+1/k

−3

−3+1/k


K ⊂

L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)

L(12, 5)L(10, 3)

and φ ∼=

τxτy

τ2x τy

τ3x τy

τ4x τy

τ5x τy

.



∆ > 1


−1

−6+1/k

−2

−4+1/k

−3

−3+1/k


K ⊂

L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)

L(12, 5)L(10, 3)

and φ ∼=

τxτy

τ2x τy

τ3x τy

τ4x τy

τ5x τy

.





m


y τmx τ−1



φ = τxτy and ∆ = | − 6k + 1| for k ∈ Z

φ = τ2x τy and ∆ = | − 4k + 1| for k ∈ Z

φ = τ3x τy and ∆ = | − 3k + 1| for k ∈ Z

φ = τ4x τy and ∆ = 1, 3

φ = τ5x τy and ∆ = 1, 2




m


y τmx τ−1



φ = τxτy and ∆ = | − 6k + 1| for k ∈ Z

φ = τ2x τy and ∆ = | − 4k + 1| for k ∈ Z

φ = τ3x τy and ∆ = | − 3k + 1| for k ∈ Z

φ = τ4x τy and ∆ = 1, 3

φ = τ5x τy and ∆ = 1, 2


Documents

Kenneth L. Baker [email protected]/Zacatecas/zacatecas... · 2007. 5. 31. · Kenneth L. Baker (GaTech) Lens space ﬁllings of OPT–bundles AMS-SMM ’07 10 / 26. Problem Extend