Transverse knots and Khovanov homology

Abstract

We define an invariant of transverse links in (S 3 , ξ std ) as a distinguished element of the Khovanov homology of the link. The quantum grading of this invariant is the self-linking number of the link. For knots, this gives a bound on the self-linking number in terms of Rasmussen's invariant s(K). We prove that our invariant vanishes for transverse knot stabilizations, and that it is non-zero for quasipositive braids. We also discuss a connection to Heegaard Floer invariants.

Khovanov homology, sutured Floer homology and annular links

“…x L/ (where Kh is a version of Khovanov homology for links in product manifolds defined by Asaeda, Przytycki and Sikora [1]) to (a variant of) the knot Floer homology of z B †.S 3 ; L/, where z B is the preimage of B in †.S 3 ; L/. 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.…”

“…Heegaard Floer homology as studied by Ozsváth and Szabó [19] and Khovanov homology [10] have transformed the landscape of low-dimensional topology in the past decade, generating a wealth of applications, most notably to questions in knot concordance (cf Ozsváth and Szabó [16] and Rasmussen [27; 26]), Dehn surgery (cf Ozsváth and Szabó [21] and Watson [29]) and contact geometry (cf Ozsváth and Szabó [20] and Plamenevskaya [25]). The philosophies underlying the theories' constructions are quite different, yet there are intriguing connections between the two.…”

Floer theory and low dimensional topology

“…Ozsváth-Szabó also define knot invariants in [26] and use them in [27] to give a new obstruction for a knot to have unknotting number one. Heegaard Floer theory has many other interesting consequences for knot theory; see for example recent papers by Ekterkary [5], Livingston-Naik [18], Plamenevskaya [29], and Rasmussen [30].…”

Braids, transversal links and the Khovanov-Rozansky Theory

“…In [11], O. Plamenevskaya constructed an invariant ψ of transversal links in standard contact S 3 using the original Khovanov homology H defined in [6]. We generalize her construction, and, for each n ≥ 2, define an invariant ψ n of transversal links in standard contact S 3 using H n .…”