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John A. Baldwin and Olga Plamenevskaya- Khovanov Homology, Open Books and Tight Contact Structures
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a r X i v : 0 8 0 8 . 2 3 3 6 v 2 [ m a t h . G T ] 2 6 S e p 2 0 0 8 KHOV ANOV HOMOLOGY, OPEN BOOKS, AND TIGHT CONTACT STRUCTURES JOHN A. BALDWIN AND OLGA PLAMENEVSKAYA Abstract. We define the reduced Khovanov homology of an open book (S, φ), and we iden- tify a disti nguis hed “conta ct ele men t” in this group whic h may be used to esta blis h the tightness or non-fillability of contact structures compatible with (S, φ). Our constr uct ion generalizes the relationship between the reduced Khovanov homology of a link and the Hee- gaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for severa l contact structure s whic h are not Stei n-fill able . Last ly , we in- vestigate a comultiplication structure on the reduced Khovanov homology of an open book which parallels the comultiplication on Heegaard Floer homology defined in [5]. 1. Introduction The goal of this paper is to demonstrate how Khovanov homology and related ideas may be used to combinatorially establish the tightness or non-fillability of certain contact structures. Let S be a compact, oriented surface with boundary, and let φ be a composition of Dehn twis ts around homotopically non-trivial curves in S . The abstract open book (S, φ) corresponds to a contact 3-manifold, which we denote by ( M S,φ , ξ S,φ ) [10, 37]. In this paper, we us e the link surgeries spectral seque nce mac hiner y of Ozsv´ ath and Szab´ o [30] to define a filtered chain complex (C (S, φ), D) whose homology is isomorphic to HF (−M S,φ ) (we work with Z 2 coefficie nts througho ut). W e then define the reduced Khovanov homology of the open book (S, φ) to be the E 2 term of the spectral sequence associated to this filtered complex, and we identify an element ψ(S, φ) ∈ Kh (S, φ) which is related to the Ozsv´ath-Szab´ o contact invariant c(S, φ) ∈ HF (−M S,φ ) via this spectral sequence. Let S k,r denote the genus k surface with r boundary components. By Giroux’s correspon- dence [10], every contact 3-manifold is compatible with an open book of the form (S k,1 , φ). Moreover, any boundary-fixing diffeomorphism of S k,1 is isotopic (rel. ∂S k, 1 ) to a composi- tion of Dehn twists around the curves α 0 ,..., α 2k depicted in Figure 1 [15]. We show that Kh (S k, 1 , φ) and ψ(S k,1 , φ) are combinatorially computable when φ is such a composition. When the contact manifold ( M S,φ , ξ S,φ ) is the branched double cover of a transverse link, our construction specializes to the setup of [32]. Indeed, let w = w(σ 1 , σ −1 1 ,..., σ 2k , σ −1 2k ) be a word in the elementary generators (and their inverses) of the braid group on 2k +1 stran ds, and let L w denote the closure of the braid specified by w. We may think of L w as a transverse link in standard contact structure ξ st on S 3 , and lift ξ st to a contact structure ξ Lw on the double cover Σ(L w ) of S 3 branched along L w . The contact structure ξ Lw is compatible with a natural open book decomposition (S k,1 , φ w ) of Σ(L w ), where φ w = w(D α 1 ,D −1 α 1 ,...,D α 2k ,D −1 α 2k ) (here, D γ The first author was partially supported by an NSF Postdoctoral Fellowship. 1
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