Legendrian knots, transverse knots and combinatorial Floer homology

By Peter S. Ozsv'ath, Zolt'an Szab'o, Dylan Thurston. 2006

142 citations recorded by the source corpus.

Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.

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

View Paper: https://www.semanticscholar.org/paper/22fa91335a94c9432c2c7069d5286ca1ce31e8de

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Topologically Trivial Legendrian Knots (2008, background): “binatorial version of knot Floer homology [ OST ].”
Cables of thin knots and bordered Heegaard Floer homology (2009, background): “ant τ(K) arising from the Alexander ﬁltration on CFd(S3). Alternatively, τ can be deﬁned in terms of the associated graded object HFK−(K) by τ(K) = −max{s|∀d ≥ 0,UdHFK−(K,s) 6= 0 }. (see Lemma A.2 of [15]). We do not need to fully compute the homology of gCFK−(Kp,pn+1). It is enough to observe that ”
Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology (2007, methodology): “O. Thus one easily checks, by a calculation analogous to that detailed in the proof of Lemma 6.3 of [ OST06 ], that Mf(exO) = −(N −1), where N is the grid number (see also Lemma 3.2 in [MOS06]).”
A survey of Heegaard Floer homology (2013, background): “t Floer homology HFK{pS3,Kq, where the input data is a grid diagram for K. The latter led to an invariant of Legendrian and transverse knots in contact 3-manifolds, cf. Ozsv´ath, Szabo´, and Thurston [81]. The hat version of HF is considerably simpler to compute than the other ﬂavors. It took Manole”
Heegard Floer invariants of Legendrian knots in contact three-manifolds (2008, background): “The reformulation of knot Floer homology (introduced originally in [35, 39]) through grid diagrams [26, 27] provided not only a combinatorial way of computing the knot Floer homology groups of knots in S3, but also showed a natural way of defining invariants of Legendrian and transverse knots in the”
A cube of resolutions for knot Floer homology (2007, background): “Λ is a cycle since each rectangle leaving it must contain one or two Xs (compare [14]).”
On the equivalence of Legendrian and transverse invariants in knot Floer homology (2011, methodology): “Since the class θ(K) is always nonzero [OSzT08], the same is true of [x4].”
Contact surgeries and the transverse invariant in knot Floer homology (2008, background): “−Y,L) into HFK([ −Y,L); the image of the invariant L(L) under this reduction is denoted by Lb(L). 1 The motivation for the Legendrian and transverse invariants comes from the diﬀerent construction in [17], which gives an invariant for Legendrian and transverse knots in the standard contact three-sph”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “One deﬁnition (cf. [OST08]) can be given in terms of the −∗ ( K, s ) , U j x 6 = 0 for all j ≥ 0 } .”