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arXiv · 0708.0448

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

Matthew Hedden
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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.

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Matthew Hedden. 2007-08-05. An Ozsvath-Szabo Floer homology invariant of knots in a contact manifold. https://arxiv.org/abs/0708.0448

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