On a Relation Between the Self-Linking Number and the Braid Index of Closed Braids in Open Books

On a relation between the self-linking number and the braid index of closed braids in open books

Tetsuya Ito (RIMS, Kyoto University)

2015 Sep 7 Braids, Conﬁguration Spaces, and Quantum Topology

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 1 / 22 Notations, conventions ⟨ ⟩ σi σj σi = σi σj σi , |i − j| = 1 Bn = σ1, . . . , σn−1 : braid group σi σj = σj σi , |i − j| > 1 We use the following notations.

▶ e : Bn → Z: exponent sum ▶ αb: the closure of a braid α ▶ n(α) (or, n(αb)): the number of strands of a (closed) braid α Def (i.e. n(α) = n if α ∈ Bn) ▶ b(K) = braid index of a knot K Def= min{n(α) | αb = K}.

Convention Every 3-manifold is closed and oriented (unless otherwise speciﬁed)

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 2 / 22 Simple supporting evidence from quantum topology For closed 3-braid αb, the Jones polynomial is given by

e/2 −1 e e/2 −1 Vαb(t) = t (t + 1 + t + t − t (t + 1 + t)∆αb(t)

⇒ except a few special case, e = e(α) must be an invariant of αb.

Jones’ conjecture

Inspired by Hecke algebra representation formula of HOMFLY polynomial V. Jones raised the following question: Jones’ conjecture (’87) For a knot K, e = e(β) is a knot invariant if n(β) = b(K) (i.e. β is a minimal closed braid representative)

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 3 / 22 Jones’ conjecture

Simple supporting evidence from quantum topology For closed 3-braid αb, the Jones polynomial is given by

e/2 −1 e e/2 −1 Vαb(t) = t (t + 1 + t + t − t (t + 1 + t)∆αb(t)

⇒ except a few special case, e = e(α) must be an invariant of αb.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 3 / 22 Recently, the conjecture is proved.

Theorem (Dynnikov-Prasolov ’13, LaFountain-Menasco ’14) The Jones-Kawamuro conjecture is true.

Jones-Kawamuro conjecture

Later, K. Kawamuro proposed a generalization of Jones’ conjecture for non-minimal closed braid representatives. Jones-Kawamuro conjecture (’06) If two closed braids αb and βb are isotopic to the same knot/link K,

|e(α) − e(β)| ≤ n(α) + n(β) − 2b(K)

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 4 / 22 Jones-Kawamuro conjecture

|e(α) − e(β)| ≤ n(α) + n(β) − 2b(K)

Recently, the conjecture is proved.

Theorem (Dynnikov-Prasolov ’13, LaFountain-Menasco ’14) The Jones-Kawamuro conjecture is true.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 4 / 22 Contact topology point of view

Surprisingly, Jones-Kawamuro conjecture, although ﬁrst inspired from quantum topology, turns out to be related to contact topology of 3-manifolds, where the braid group also plays a crucial role.

Deﬁnition A contact structure on a 3-manifold M is a plane ﬁeld of the form

ξ = Ker α, α ∧ dα > 0 (α : 1-form on M).

A 3-manifold M equipped with a contact structure (M, ξ) is called a contact 3-manifold.

c.f. Foliation ξ is a (tangent plane ﬁeld of) foliation ⇐⇒ α ∧ dα = 0.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 5 / 22 Example: Standard contact structure (r, θ, z): Cylindrical coordinate of R3 ⊂ S3 = R3 ∪ {∞} 2 ξstd = Ker(dz + r dθ)

(Picture bollowed from P. Massot’s web page)

Fact (Darboux’s theorem) Every contact structure is locally isomorphic to the standard contact 3 ∼ 3 structure (R , ξstd ) = (R , Ker(dz + xdy)).

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 6 / 22 Transverse knot and self-linking number

Deﬁnition An oriented knot K in contact 3-manifold (M, ξ) is a transverse knot if K positively transverse ξp at every p ∈ K ⊂ M.

Deﬁnition For a null-homologous transverse knot K ⊂ (M, ξ) and its Seifert surface Σ, the relative euler number

2 sl(K; Σ) = −⟨e(ξ|Σ), [Σ])⟩ (e(ξ) ∈ H (M; Z) : Euler class of ξ).

is called the self-linking number. This is an invariant of transverse knot, depending [Σ] ∈ H2(M, K). (So it does not depend on Σ if M is an integral homology sphere.)

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 7 / 22 How does Bennequin distinguish an “exotic” contact structure ? – he used (closed) braids!! Bennequin’s observation A closed braid αb represents a transverse knot/link in 3 2 (S , ξstd = Ker(dz + r dθ)).

Proof: When r is large, ξ ∼ Ker(dθ = 0).

Bennequin’s work

Theorem (Bennequin ’83) – Birth of contact topology 3 For a contact structure ξot = Ker(cos(r)dz + r sin(r)dθ) of S ,

1. ξot is homotopic to ξstd as plane ﬁelds.

2. ξot is not isotopic to ξstd as contact structures.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 8 / 22 Bennequin’s work

Theorem (Bennequin ’83) – Birth of contact topology 3 For a contact structure ξot = Ker(cos(r)dz + r sin(r)dθ) of S ,

1. ξot is homotopic to ξstd as plane ﬁelds.

2. ξot is not isotopic to ξstd as contact structures.

How does Bennequin distinguish an “exotic” contact structure ? – he used (closed) braids!! Bennequin’s observation A closed braid αb represents a transverse knot/link in 3 2 (S , ξstd = Ker(dz + r dθ)).

Proof: When r is large, ξ ∼ Ker(dθ = 0).

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 8 / 22 On the other hand, for ξot there exists a transverse unknot K with sl(K) = 1. =⇒ ξstd and ξot must be diﬀerent as contact structures.

Bennequin’s work (continued)

Bennequin’s formula The self-linking number of a closed braid αb (viewed as a transverse knot in 3 (S , ξstd )) is given by

sl(αb) = −n(α) + e(α)

Bennequin’s inequality 3 For any transverse knot K in (S , ξstd ),

sl(αb) ≤ 2g(K) − 1

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 9 / 22 Bennequin’s work (continued)

Bennequin’s formula The self-linking number of a closed braid αb (viewed as a transverse knot in 3 (S , ξstd )) is given by

sl(αb) = −n(α) + e(α)

Bennequin’s inequality 3 For any transverse knot K in (S , ξstd ),

sl(αb) ≤ 2g(K) − 1

On the other hand, for ξot there exists a transverse unknot K with sl(K) = 1. =⇒ ξstd and ξot must be diﬀerent as contact structures.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 9 / 22 Overtwisted contact structure

(r, θ, z): Cylindrical coordinate of R3 ⊂ S3 = R3 ∪ {∞} ξot = Ker(cos(r)dz + r sin(r)dθ)

(Picture bollowed from P. Massot’s web page)

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 10 / 22 Overtwisted contact structure

(r, θ, z): Cylindrical coordinate of R3 ⊂ S3 = R3 ∪ {∞} ξot = Ker(cos(r)dz + r sin(r)dθ)

(Picture bollowed from P. Massot’s web page)

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 10 / 22 Contact topology point of view

In terms of Bennequin’s formula sl(αb) = −n(α) + e(α), Jones-Kawamuro conjecture is written as Self-linking number version of Jones-Kawamuro conjecture If two closed braids αb and βb are isotopic to the same topological knot/link K, then |sl(αb) − sl(βb)| ≤ 2 max{n(α), n(β)} − 2b(K)

=⇒ Jones-Kawamuro conjecture (inspired from Quantum topology) provides an interaction between

self-linking number, transverse knot invariant and closed braid representatives of a topological knot.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 11 / 22 Motivating question

A relation between closed braids and transverse knot (contact 3-manifolds) can be generalized: ▶ Open book decomposition =⇒ presentation of contact 3-manifolds ▶ Closed braids in open book =⇒ presentation of transverse knots

Motivating question Generalize the Jones-Kawamuro conjecture for general contact 3-manifolds

Remark ▶ There is a self-linking number formula of closed braids in general open book (I-Kawamuro ’14) ▶ Bennequin’s inequality is generalized for tight contact 3-manifolds (Eliashberg ’89)

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 12 / 22 Open book decomposition

▶ S: Oriented compact surface with non-empty boundary ▶ Diﬀ Aut(S, ∂S) = {ϕ : S → S | ϕ|∂S = id}/Diﬀeotopy = MCG(S) = Mapping class group of S

Convention We will often confuse ϕ ∈ Aut(S, ∂S) with diﬀeomorphism ϕ : S → S.

Deﬁnition A pair (S, ϕ) is called an (abstract) open book.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 13 / 22 Open book manifold For an open book (S, ϕ), deﬁne ∪ ⊔ Def × ∼ 1 × 2 M(S,ϕ) = |S [0, 1]/(x,{z1) (ϕ(x), 0)} (S D ) #∂S Mapping torus | {z }

Solid tori

fg ¢ ¾

¢ fg

= Ë ¢ fØg

¢ fg

An n-braid β of a surface S gives rise to a knot/link in M(S,ϕ).

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 14 / 22 Transverse Markov Theorem (Pavalescu,Mitsumatsu-Mori) There is a one-to-one correspondence between two sets

{(Closed) braids }/positive stabilization (+ conjugacy) and Transverse links in (M(S,ϕ), ξ(S,ϕ))}/Transverse isotopy

Giroux correspondence and transverse Markov Theorem

Giroux correspondence (Giroux, 2002) There is a one-to-one correspondence between two sets

{Open books}/stabilization (+ conjugacy) and {Contact 3-manifolds}/contactomorphism

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 15 / 22 Giroux correspondence and transverse Markov Theorem

Giroux correspondence (Giroux, 2002) There is a one-to-one correspondence between two sets

{Open books}/stabilization (+ conjugacy) and {Contact 3-manifolds}/contactomorphism

Transverse Markov Theorem (Pavalescu,Mitsumatsu-Mori) There is a one-to-one correspondence between two sets

{(Closed) braids }/positive stabilization (+ conjugacy) and Transverse links in (M(S,ϕ), ξ(S,ϕ))}/Transverse isotopy

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 15 / 22 Unfortunately... Bad news 2 3 Unless (S, ϕ) = (D , id) (even for open book decomposition of (S , ξstd ) !) there are counterexamples of generalized Jones-Kawamuro conjecture (for all (null homologous) knots – even for unknot!)

Naive expectation

Now a naive generalization of Jones-Kawamuro conjecture is: Generalized Jones-Kawamuro conjecture ? b Let α,b β be a closed braid in M(S,ϕ). Assume that 1. αb and βb are topologically isotopic (to the same knot K) Then |sl(αb) − sl(βb)| ≤ 2 max{n(αb), n(βb)} − 2b(K)

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 16 / 22 Naive expectation

Unfortunately... Bad news 2 3 Unless (S, ϕ) = (D , id) (even for open book decomposition of (S , ξstd ) !) there are counterexamples of generalized Jones-Kawamuro conjecture (for all (null homologous) knots – even for unknot!)

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 16 / 22 Main Theorem

To get correct generalization, we need several assumptions and modiﬁcations. The correct generalization is: Theorem (I. Generalized Jones-Kawamuro conjecture) b Let α,b β be a closed braid in M(S,ϕ). Assume that 1. αb and βb are C-topologically isotopic (to a knot K). 2. S is planar. 3. c(ϕ, C) > 1, where c(ϕ, C) is the Fractional Dehn twist coeﬃcient of ϕ along C. for some connected component C of the binding B. Then b b |sl(αb) − sl(β)| ≤ 2 max{n(αb), n(β)} − 2bC (K)

The key point is: we need to consider the notion of C-topologically isotopic.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 17 / 22 Fractional Dehn twsit coeﬃcients

C: Connected component of ∂S. The fractional Dehn twist coeﬃcient (FDTC, in short) is a map

c(∗, C): MCG(S) → Q

introduced by Honda-Kazez-Matic in ’08.

c(ϕ, C) represents how many times ϕ ∈ MCG(S) twists the boundary C.

´ µ = ¿

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 18 / 22 C-Topological isotopy

Deﬁnition Fix one connected component of the binding C ⊂ B = ∂S. We say two knots in M − B are C-topologically isotopic if they are isotopic in M − (B − C). (i.e. K can across C, but cannot across other bindings.)

Deﬁnition For a knot K ⊂ M − B, deﬁne the C-braid index of K by

Def bC (K) = min{n(α) | αb is C-topologically isotopic to K}.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 19 / 22 Main Theorem, again

Theorem (I. Generalized Jones-Kawamuro conjecture) b Let α,b β be a closed braid in M(S,ϕ). Assume that 1. αb and βb are C-topologically isotopic 2. S is planar. 3. c(ϕ, C) > 1, where c(ϕ, C) is the Fractional Dehn twist coeﬃcient of ϕ along C. for some connected component C of the binding B. Then b b |sl(αb) − sl(β)| ≤ 2 max{n(αb), n(β)} − 2bC (K)

Roughly speaking: The self-linking number is controlled when we look at particular binding C.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 20 / 22 Then the assumptions of main theorem (C-top isotopic, planarity, FDTC) allows us to modify embedded annulus A in “standard form” which proves desired inequality.

Idea of proof

The proof uses open book foliation techinque (singular foliation induced by intersection with pages), developed by I-Kawamuro. The key assertion is: Annulus Lemma (LaFountain-Menasco for S 3, general case by I.) b If αb and β are topologically isotopic, then there exists closed braids αb+ b and β− such that ▶ b αb+ and β− cobound an embedded annulus A. ▶ αb+ is a positive stabilization of αb. b b ▶ β− is a negative stabilization of β.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 21 / 22 Idea of proof

Then the assumptions of main theorem (C-top isotopic, planarity, FDTC) allows us to modify embedded annulus A in “standard form” which proves desired inequality.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 21 / 22 Refereces This talk is based on my preprint

▶ T. Ito, On a relation between the self-linking number and the braid index of closed braids in open books, arXiv:1501.07314. The original Jones-Kawamuro conjecture are proven in ▶ I. Dynnikov and M. Prasolov, Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions, Trans. Moscow Math. Soc. (2013), 97–144. ▶ D. LaFountain and W. Menasco, Embedded annuli and Jones’ conjecture, Algebr. Geom. Topol. 14 (2014), 3589–3601. For origins of the conjecture (and relations to quantum invariants), ▶ V. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987), 335-388. ▶ K. Kawamuro, The algebraic crossing number and the braid index of knots and links, Algebr. Geom. Topol. 6 (2006), 2313–2350.

Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 22 / 22