An Ozsváth–Szabó Floer homology invariant of knots in a contact manifold

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 Ozsváth-Szabó 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 S 3 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 Ozsváth-Szabó invariants, then any fibered knot can realize the classical Eliashberg-Bennequin bound in at most one of these contact structures.

Some 3-dimensional transverse $\mathbb{C}$-links (Constructions of higher-dimensional $\mathbb{C}$-links, I)

“…Theorem (Giroux [33]; see also Giroux and Goodman [34]) Every contact structure on S 3 is ambient isotopic to a contact structure ξ 2.27 Theorem ( [80]; see also Hedden [41]) A fibered link L = L(b) in S 3 is strongly quasipositive iff up to ambient isotopy b is compatible with the standard, contact structure ξ 0 on S 3 .…”

Contact homology of orbit complements and implied existence

“…Consider now specifically a tight fibered hyperbolic knot realizing the Thurston Bennequin bound in the tight S 3 . By [Hed08], tight fibered knots realizing the Thurston-Bennequin bound are determined by the topological knot type up to transverse isotopy, so we may assume without loss of generality that such a knot is in fact the binding of an open book decomposition supporting the tight S 3 with (pseudo-Anosov) monodromy map φ. By definition, the open book supports the tight contact structure, so the kernel of the constructed contact form (which supports this open book) is the tight contact structure on S 3 (up to isotopy).…”

A bound for rational Thurston–Bennequin invariants

“…In this paper, we introduce an invariant τ * c(ξ) (Y, K, F ) for an rationally null-homologous knot K, which generalizes Hedden's definition [9]. Our main theorem proves that this invariant gives an upper bound for the sum of the rational Thurston-Bennequin invariant and the rational rotation number of all Legendrian representatives of K. Theorem 1.1.…”