An Ozsvath-Szabo Floer homology invariant of knots in a contact manifold

TL;DR

This paper introduces a new knot invariant in contact three-manifolds derived from Floer homology, which bounds Legendrian knot invariants and helps identify tight contact structures.

Contribution

It defines a novel invariant using Ozsvath-Szabo Floer homology that extends knot invariants to contact manifolds and provides criteria for tightness and Legendrian bounds.

Findings

Constructs prime knots with negative Thurston-Bennequin invariant in contact manifolds.

Provides a criterion for open books to induce tight contact structures.

Shows that distinct contact structures with non-vanishing invariants restrict Legendrian realizations.

Abstract

Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston-Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than the three-sphere with negative maximal Thurston-Bennequin invariant. Perhaps more interesting, our invariant provides a criterion for an open book to induce a tight contact structure. A corollary is that if a manifold possesses contact structures with distinct non-vanishing Ozsvath-Szabo invariants, then any fibered knot can realize the classical Eliashberg-Bennequin bound in at most one of these contact structures.