Khovanov homology, open books, and tight contact structures

Abstract

We define the reduced Khovanov homology of an open book (S, φ), and identify a distinguished "contact element" in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S, φ). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology of an open book which parallels the comultiplication on Heegaard Floer homology defined in Baldwin (2008) [4].

Khovanov homology, sutured Floer homology and annular links

“…This allowed him to establish a relationship, first conjectured in [25], between Plamenevskaya's transverse invariant [25] and Ozsváth and Szabó's contact invariant [20]. Baldwin and Plamenevskaya [4] used (an extension of) this relationship to establish the tightness of a number of non-Stein-fillable contact structures.…”

On the Naturality of the Spectral Sequence from Khovanov Homology to Heegaard Floer Homology

“…(1) trivial inclusion (see Figure 1), (2) horizontal stacking (see Figure 2), and (3) vertical cutting (see Figure 3). In particular, let F (T ) := X(L T ) (resp., F (L) := X(L L )) denote the filtered chain complex, described above and in Notation 2.7, associated to the balanced tangle T ⊂ D × I (resp., link L ⊂ A × I).…”

“…• a proof, in [4], that Khovanov's categorification, [12], of the reduced, n-colored Jones polynomial detects the unknot whenever n ≥ 2, as well as • a new method, due to Baldwin-Plamenevskaya [2], for establishing the tightness of certain contact structures. In [5], we recast [18], [19], and [4] as specific instances of a broader relationship between Khovanov-and Heegaard Floer-type homology theories, using a version of Heegaard Floer homology for sutured manifolds developed by Juhász in [7].…”

Contact structures on plumbed 3-manifolds

“…The invariant c(ξ) can be calculated combinatorially as shown in [25] and [1]. However, for the present applications the usage of the c + is essential.…”