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

Transcription

1 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

2 Notations, conventions B n = σ 1,..., σ n 1 σ i σ j σ i = σ i σ j σ i, i j = 1 σ i σ j = σ j σ i, i j > 1 We use the following notations. e : B n Z: exponent sum α: the closure of a braid α : braid group n(α) (or, n( α)): the number of strands of a (closed) braid α (i.e. n(α) Def = n if α B n ) b(k) = braid index of a knot K Def = min{n(α) α = 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

3 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

4 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 α, the Jones polynomial is given by V α (t) = t e/2 (t t + t e t e/2 (t t) α (t) except a few special case, e = e(α) must be an invariant of α. Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 3 / 22

5 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 α and β 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

6 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 α and β are isotopic to the same knot/link K, Recently, the conjecture is proved. e(α) e(β) n(α) + n(β) 2b(K) 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

7 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

8 Example: Standard contact structure (r, θ, z): Cylindrical coordinate of R 3 S 3 = R 3 { } ξ std = Ker(dz + r 2 dθ) Fact (Darboux s theorem) (Picture bollowed from P. Massot s web page) Every contact structure is locally isomorphic to the standard contact structure (R 3, ξ std ) = (R 3, Ker(dz + xdy)). Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 6 / 22

9 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 ξ p at every p K M. Definition For a null-homologous transverse knot K (M, ξ) and its Seifert surface Σ, the relative euler number sl(k; Σ) = e(ξ Σ ), [Σ]) (e(ξ) H 2 (M; Z) : Euler class of ξ). is called the self-linking number. This is an invariant of transverse knot, depending [Σ] H 2 (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

10 Bennequin s work Theorem (Bennequin 83) Birth of contact topology For a contact structure ξ ot = Ker(cos(r)dz + r sin(r)dθ) of S 3, 1. ξ ot is homotopic to ξ std as plane fields. 2. ξ ot is not isotopic to ξ std as contact structures. Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 8 / 22

11 Bennequin s work Theorem (Bennequin 83) Birth of contact topology For a contact structure ξ ot = Ker(cos(r)dz + r sin(r)dθ) of S 3, 1. ξ ot is homotopic to ξ std as plane fields. 2. ξ ot is not isotopic to ξ std as contact structures. How does Bennequin distinguish an exotic contact structure? Bennequin s observation he used (closed) braids!! A closed braid α represents a transverse knot/link in (S 3, ξ std = Ker(dz + r 2 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

12 Bennequin s work (continued) Bennequin s formula The self-linking number of a closed braid α (viewed as a transverse knot in (S 3, ξ std )) is given by Bennequin s inequality sl( α) = n(α) + e(α) For any transverse knot K in (S 3, ξ std ), sl( α) 2g(K) 1 Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 9 / 22

13 Bennequin s work (continued) Bennequin s formula The self-linking number of a closed braid α (viewed as a transverse knot in (S 3, ξ std )) is given by Bennequin s inequality sl( α) = n(α) + e(α) For any transverse knot K in (S 3, ξ std ), sl( α) 2g(K) 1 On the other hand, for ξ ot there exists a transverse unknot K with sl(k) = 1. = ξ std and ξ ot must be different as contact structures. Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 9 / 22

14 Overtwisted contact structure (r, θ, z): Cylindrical coordinate of R 3 S 3 = R 3 { } ξ ot = 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

15 Overtwisted contact structure (r, θ, z): Cylindrical coordinate of R 3 S 3 = R 3 { } ξ ot = 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

16 Contact topology point of view In terms of Bennequin s formula sl( α) = n(α) + e(α), Jones-Kawamuro conjecture is written as Self-linking number version of Jones-Kawamuro conjecture If two closed braids α and β are isotopic to the same topological knot/link K, then sl( α) sl( β) 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

17 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

18 Open book decomposition S: Oriented compact surface with non-empty boundary Aut(S, S) = {ϕ : S Diff 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

19 Open book manifold For an open book (S, ϕ), define M (S,ϕ) Def = S [0, 1]/(x, 1) (ϕ(x), 0) (S 1 D 2 ) }{{} # S Mapping torus }{{} Solid tori Ô Ë ½ Ô ¾ ˵ ÓÙÒ Ò Ò Ë ½ Ë ¼ Ë Ë Ø Ë Ø Ë ½ 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

20 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

21 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

22 Naive expectation Now a naive generalization of Jones-Kawamuro conjecture is: Generalized Jones-Kawamuro conjecture? Let α, β be a closed braid in M (S,ϕ). Assume that 1. α and β are topologically isotopic (to the same knot K) Then sl( α) sl( β) 2 max{n( α), n( β)} 2b(K) Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 16 / 22

23 Naive expectation Now a naive generalization of Jones-Kawamuro conjecture is: Generalized Jones-Kawamuro conjecture? Let α, β be a closed braid in M (S,ϕ). Assume that 1. α and β are topologically isotopic (to the same knot K) Then Unfortunately... Bad news sl( α) sl( β) 2 max{n( α), n( β)} 2b(K) Unless (S, ϕ) = (D 2, id) (even for open book decomposition of (S 3, ξ std )!) 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

24 Main Theorem To get correct generalization, we need several assumptions and modifications. The correct generalization is: Theorem (I. Generalized Jones-Kawamuro conjecture) Let α, β be a closed braid in M (S,ϕ). Assume that 1. α and β 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 sl( α) sl( β) 2 max{n( α), n( β)} 2b C (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

25 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 ¾ Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 18 / 22

26 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 b C (K) Def = min{n(α) α is C-topologically isotopic to K}. Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 19 / 22

27 Main Theorem, again Theorem (I. Generalized Jones-Kawamuro conjecture) Let α, β be a closed braid in M (S,ϕ). Assume that 1. α and β 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 sl( α) sl( β) 2 max{n( α), n( β)} 2b C (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

28 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.) If α and β are topologically isotopic, then there exists closed braids α + and β such that α + and β cobound an embedded annulus A. α + is a positive stabilization of α. β is a negative stabilization of β. Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 21 / 22

29 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.) If α and β are topologically isotopic, then there exists closed braids α + and β such that α + and β cobound an embedded annulus A. α + is a positive stabilization of α. β is a negative stabilization of β. 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

30 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: 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), D. LaFountain and W. Menasco, Embedded annuli and Jones conjecture, Algebr. Geom. Topol. 14 (2014), 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), K. Kawamuro, The algebraic crossing number and the braid index of knots and links, Algebr. Geom. Topol. 6 (2006), Tetsuya Ito (RIMS, Kyoto University) Self-linking number and braid index 2015 Sep 7 22 / 22