Bounds for the Thurston–Bennequin number from Floer homology

By O. Plamenevskaya. 2003

89 citations recorded by the source corpus.

Using a knot concordance invariant from the Heegaard Floer theory of Ozsvath and Szabo, we obtain new bounds for the Thurston- Bennequin and rotation numbers of Legendrian knots in S 3 . We also apply these bounds to calculate the knot concordance invariant for certain knots. AMS Classification 57R17, 57M27

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b33bcaf61e7bd75cb4c2a8c45268eba53d729e5f

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Citations in the corpus

A concordance invariant from the Floer homology of double branched covers (2005, background): “Since TB(K) ≤ 2τ(K) − 1 by the work of Plamenevskaya [31], Rudolph’s result is a consequence of ours.”
A Legendrian Thurston-Bennequin bound from Khovanov homology (2005, background): “• tb(K) + |r(K)| ≤ 2g(K) − 1, Bennequin’s original result [2], where g is the genus of K ; • tb(K) + |r(K)| ≤ 2g4(K) − 1 [23], where g4 is the slice genus of K ; • tb(K) + |r(K)| ≤ min-dega HK(a, z) − 1 [8, 16], where HK is the HOMFLYPT polynomial and min-dega is the minimum degree in the variable a”
On knot Floer homology and cabling (2004, background): “≥ 0. Then n ≥ −2 τ(K) p−1 + 1 p . ON KNOT FLOER HOMOLOGY AND CABLING II 19 This is relevant in light of the connection between τ(K) and complex curves. Suppose that K is a C-knot. Then results of [31, 13, 36] show that τ(K) = g 4(K). In particular, τ(K) ≥ 0. Combining this fact with the corollary ”
NOTIONS OF POSITIVITY AND THE OZSVÁTH–SZABÓ CONCORDANCE INVARIANT (2005, background): “(Plamenevskaya [28]) Suppose K is quasipositive.”
On Knot Floer Homology and Cabling: 2 (2008, background): “Then results of [31, 13, 36] show that τ(K) = g4(K).”
Transverse knots and Khovanov homology (2004, methodology): “rs of Legendrian knots (Legendrian and transverse knots are related by push-oﬀs [Et]), for an 14 OLGA PLAMENEVSKAYA alternating Legendrian knot Kwe have tb(K)+|r(K)| ≤ σ−1. This bound was obtained in [Pl1] via Heegaard Floer homology techniques. Indeed, it is a special case of the inequality tb(K)+|”
The Ozsváth-Szabó and Rasmussen Concordance Invariants are not Equal (2005, methodology): “This follows from work of Plamenevskaya [23] for τ and from Plamenevskaya [24] and Shumakovitch [30] for s.”
Transverse knots distinguished by knot Floer homology (2007, background): “We wish to thank John Baldwin, Marc Culler, John Etnyre, Hiroshi Matsuda, Olga Plamenevskaya, Jacob Rasmussen, and Zoltán Szabó for interesting conversations during the course of this work.”
Grid Homology for Knots and Links (2015, methodology): “These bounds, stated below, were first proved by Plamenevskaya [187] using the contact invariant in Heegaard Floer homology [175].”