Heegard Floer invariants of Legendrian knots in contact three-manifolds

Abstract

Abstract. We define invariants of null-homologous Legendrian and transverse knots in contact 3-manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non-loose knots in overtwisted 3-spheres. Moreover, we apply the invariants to find transversely non-simple knot types in many overtwisted contact 3-manifolds.

Dehn twists in Heegaard Floer homology

“…This is already stated in HondaKazez-Matić [13] and Lisca-Ozsváth-Stipsicz-Szabó [15] but not proved. We wish to give a proof because this specific choice is crucial throughout this article.…”

“…This invariant is due to Lisca, Ozsváth, Stipsicz and Szabó and was defined in [15]. It is basically the contact element, but now it is interpreted as sitting in a filtered Heegaard Floer complex.…”

“…Let .P; / be an open book decomposition that is adapted to the contact structure (see Section 2.4). Recall that the contact element and the invariant defined by Lisca, Ozsváth, Stipsicz and Szabó [15] denotes a negative Dehn twist along L with respect to the orientation of P . Observe that L sits on the wrong page for our construction of the exact sequence.…”

“…The associated Dehn twists can be interpreted as contact surgeries along suitable Legendrian knots (see [15,Theorem 2.7]). Thus, using the open book decomposition we are able to find a (maybe very inefficient) contact surgery representation of .Y i ; i / which is suitable for our purposes to perform the Legendrian band sum (see the beginning of this section).…”

“…where is a closed curve in P [ h 1 that intersects the co-core of h 1 once, transversely. Fixing a homologically essential, simple closed curve ı in P we call the Giroux stabilization ı -elementary if, after a suitable isotopy, ı intersects transversely in at most one point (see [15,Definition 2.5]). Their invariance proof relies on the fact that, given a positive Giroux stabilization, one can choose a cut system a 1 ; : : : ; a n of .P; / such that the curve does not intersect any of the a i .…”