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.

Abstract. We use the Heegaard Floer theory developed by P. Ozsváth and Z. Szabó to give a new proof of a theorem of P. Lisca and G. Matić. In particular, we prove that the contact structures on Y = ∂X induced by non-homotopic Stein structures on the 4-manifold X have distinct Heegaard Floer invariants. Our examples also show that Heegaard Floer homology can distinguish between non-isotopic tight contact structures. IntroductionIn

Using a knot concordance invariant from the Heegaard Floer theory of Ozsváth and Szabó, we obtain new bounds for the ThurstonBennequin and rotation numbers of Legendrian knots in S 3 . We also apply these bounds to calculate the knot concordance invariant for certain knots.

We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendl's theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on L.p; 1/ has a unique filling, and describe fillable and nonfillable tight contact structures on certain Seifert fibered spaces. 57R17; 53D35

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].

We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic transverse knots with the same self-linking number. These pairs of knots include most of the non-transversely simple knots of Birman-Menasco and Ng-Ozsváth-Thurston. arXiv:0712.1557v1 [math.GT]

No abstract

Abstract. We show that the presence of a plastikstufe induces a certain degree of flexibility in contact manifolds of dimension 2n + 1 > 3. More precisely, we prove that every Legendrian knot whose complement contains a "nice" plastikstufe can be destabilized (and, as a consequence, is loose). As an application, it follows in certain situations that two non-isomorphic contact structures become isomorphic after connectsumming with a manifold containing a plastikstufe.