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

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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 ; ji − jj = 1 Bn = σ1; : : : ; σn−1 : braid group σi σj = σj σi ; ji − jj > 1 We use the following notations. I e : Bn ! Z: exponent sum I αb: the closure of a braid α I n(α) (or, n(αb)): the number of strands of a (closed) braid α Def (i.e. n(α) = n if α 2 Bn) I b(K) = braid index of a knot K Def= minfn(α) j αb = Kg: 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 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) 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, je(α) − e(β)j ≤ 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 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, je(α) − e(β)j ≤ 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 [ f1g 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 2 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(ξjΣ); [Σ])i (e(ξ) 2 H (M; Z) : Euler class of ξ): is called the self-linking number. This is an invariant of transverse knot, depending [Σ] 2 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 [ f1g 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 [ f1g 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 jsl(αb) − sl(βb)j ≤ 2 maxfn(α); n(β)g − 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: I Open book decomposition =) presentation of contact 3-manifolds I Closed braids in open book =) presentation of transverse knots Motivating question Generalize the Jones-Kawamuro conjecture for general contact 3-manifolds Remark I There is a self-linking number formula of closed braids in general open book (I-Kawamuro '14) I 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 I S: Oriented compact surface with non-empty boundary I Diff Aut(S;@S) = fϕ : S ! S j ϕj@S = idg=Diffeotopy = MCG(S) = Mapping class group of S Convention We will often confuse ϕ 2 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 [G Def × ∼ 1 × 2 M(S,ϕ) = |S [0; 1]=(x;{z1) (ϕ(x); 0)} (S D ) #@S Mapping torus | {z } Solid tori ÒÒ ½ fÔg ¢ Ë ´Ô ¾ Ë µ ÓÙÒ× ½ Ë Ë Ø Ë ¢ f¼g Ë = Ë ¢ fØg Ë ¢ f½g An n-braid β of a surface S gives rise to a knot/link in M(S,ϕ).
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