Knot Floer homology and rational surgeries

By Peter S. Ozsv'ath, Zolt'an Szab'o. 2005

308 citations recorded by the source corpus.

Let K be a rationally null-homologous knot in a three-manifold Y . We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot K . As an application, we express the Heegaard Floer homology of rational surgeries on Y along a null-homologous knot K in terms of the filtered homotopy type of the knot invariant for K . This has applications to Dehn surgery problems for knots in S . In a different direction, we use the techniques developed here to calculate the Heegaard Floer homology of an arbitrary Seifert fibered three-manifold with even first Betti number.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f84a9f5bb5a15db0c77845cb855eccf58cb00013

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Citations in the corpus

On Knot Floer Homology and Cabling: 2 (2008, background): “Quasipositive Seifert surfaces are those which can be obtained from parallel disks by attaching bands with a positive half twist (see Figure 1 of [11]).”
Heegaard Floer homology and integer surgeries on links (2010, unknown): “This is used as a stepping-stone to reconstruct the Heegaard Floer homology of arbitrary surgeries on a knot in [22] and [23].”
Topologically slice knots of smooth concordance order two (2012, methodology): “Although the manifolds we consider are in some respects fairly simple, in order to apply [37] it is essential to review some of the foundations.”
Involutive Heegaard Floer homology (2015, methodology): “One reason why Heegaard Floer homology is computationally tractable is because of the surgery formulas [38, 44, 45, 26] which relate it to a similar invariant for knots, knot Floer homology [38, 48].”
The reduced knot Floer complex (2013, methodology): “−1], but in this paper we will always assume coeﬃcients in F. 1 the information necessary to compute the Heegaard Floer homology of not merely S3, but also manifolds arising from Dehn surgery along K [27, 21]. Our method here will be to ignore the other Z-ﬁltration - the one which measures the expon”
Cosmetic surgeries on knots in $S^3$ (2010, methodology): “In Section 4, we use the previous results and the method in [12] to prove our main theorem.”
Floer simple manifolds and L-space intervals (2015, unknown): “In [41], Ozsváth and Szabó combined the observations above with their mapping cone for integer surgeries to express the Floer homology of any rational surgery as a mapping cone.”
A note on cabling and $L$–space surgeries (2009, methodology): “ce) surgery yields a remarkable amount of information about the Heegaard Floer invariants associated to both the knot K, and manifolds obtained by Dehn surgery on K. In particular, results of [8] and [10] allow one to compute the Heegaard Floer invariants of any Dehn surgery on an L-space knot K fro”
Grid Homology for Knots and Links (2015, methodology): “More precisely, in [181, 184], the Heegaard Floer homology groups of surgeries on K are described in terms of the filtration induced by the knot (denoted CFK−,∗ in the context of Heegaard Floer homology, generalizing the filtered complex GC− from Chapter 13).”
Heegaard Floer homology and genus one, one‐boundary component open books (2008, background): “This grading shift comes from further inspection of the Ozsváth-Szabó formula for rational surgery on knots (see [38]), combined with the fact that d(S3) = 0.”