Transverse knots distinguished by knot Floer homology

By Lenhard L. Ng, Peter S. Ozsv'ath, Dylan Thurston. 2007

72 citations recorded by the source corpus.

We use the recently-defined knot Floer homology invariant for transverse knots to show that certain pairs of transverse knots with the same self-linking number are not transversely isotopic. We also show that some of the algebraic refinements of knot Floer homology lead to refined versions of these invariants, distinguishing additional transversely non-isotopic knots with the same self-linking number.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/21005a99f7745629db37e497af7098234b825157

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Citations in the corpus

On the equivalence of Legendrian and transverse invariants in knot Floer homology (2011, result): “This computational ease underpins many of the known results regarding families of transversely non-simple knot types: Ng, Ozsváth and Thurston [26], Vértesi [41], Baldwin [3], and Khandhawit and Ng [19].”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “Knot Floer homology has further applications to contact geometry, as it allows the construction of invariants for Legendrian and transverse knots in S 3 ; see [OST08, NOT08, LOSS09, BVVV13].”
Contact surgeries and the transverse invariant in knot Floer homology (2008, background): “…this is allowed since the map on ĈF is trivial for all but finitely many Spinc-equivalence classes of maps (see [24, Theorem 3.3]); but again just as in the closed case, this is no longer the case for HFK−, and a choice of a Spinc structure s on the four-dimensional surgery cobordism is necessary.”
Combinatorial knot contact homology and transverse knots (2010, methodology): “4 23321223211 Aug(HTd 0,Z/3,2,1) = 0 10 128 12121213323 ? ? — (mirror of previous braid) — 32 L.NG Knot Grid Braid HFK? HT? HT computation m(10 132) 32233231211 X X Aug(HTd 0,Z/3,1,1) = 0 32233231121 [NOT08] Aug(HTd 0,Z/3,1,1) = 1 10 136 12123321232 ✗ X Aug(HTd 0,Z/3,2,1) = 5 23212321113 Aug(HTd 0,Z”
Computations of Heegaard-Floer knot homology (2006, background): “simple knot types [18], by Lowrance to study the relationship between knot Floer”
Heegard Floer invariants of Legendrian knots in contact three-manifolds (2008, background): “For example, the length of a homogeneous element x (that is, the minimal n for which dn(x) vanishes, where dn is the nth higher derivative in the spectral sequence associated to the filtered chain complex CFK(−Y,L)) is such a further structure, as exploited in [29].”
Grid Homology for Knots and Links (2015, methodology): “The Legendrian and transverse invariants were first defined in [185]; the applications considered here are taken from [25, 102, 157, 185].”