Heegard Floer invariants of Legendrian knots in contact three-manifolds

By P. Lisca, Peter S. Ozsv'ath, A. Stipsicz, Zolt'an Szab'o. 2008

108 citations recorded by the source corpus.

We define invariants of null–homologous Legendrian and transverse knots in contact 3–manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non–loose knots in overtwisted 3–spheres. Moreover, we apply the invariants to find transversely non–simple knot types in many overtwisted contact 3–manifolds.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f6df58dab5321e67c405532af76125f9fcdbe54f

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

On the equivalence of Legendrian and transverse invariants in knot Floer homology (2011, methodology): “Further, it is conjectured that the Legendrian invariants defined in [33] and [21] are well-behaved with respect to Lagrangian cobordism.”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “In other sources, for example in [LOSS09], the names \ CF K and CFK − are used for these associated graded.”
Khovanov homology detects the trefoils (2018, background): “ that Vela-Vick proved in [VV11] that the transverse binding Khas nonzero invariant T(K) ∈ HFK−(−Y,K), where Trefers to the transverse knot invariant deﬁned by Lisca, Ozsvath, Stipsicz, and Szabo´ in [LOSS09]. Moreover, this class lies in Alexander grading (3) (sl(K) +1)/2 = g according to [LOSS09].”
Contact surgeries and the transverse invariant in knot Floer homology (2008, methodology): “In [ 11 ] a number of explicit computations for L(L) were given by choosing the appropriate open book decompositions, and determining the homology class of the intersection point x from a direct analysis of the chain complex.”
Combinatorial knot contact homology and transverse knots (2010, methodology): “and 12n 591 are transversely nonsimple. Six of the ten knots in Proposition 1.5 were previously known to be nonsimple, using a Heegaard Floer transverse invariant (either grid-diagram [OST08] or LOSS [LOSS09]). The other four, m(7 6), m(9 44), 9 48, and 10 136, cannot be distinguished by any previou”
Grid Homology for Knots and Links (2015, background): “The present conventions are consistent with [126, 185]; see also [108, 175].”