Heegaard Floer homology and alternating knots.

By Peter S. Ozsv'ath, Zolt'an Szab'o. 2002

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In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y , which is closely related to the Heegaard Floer homology of Y . In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (generalizing a result of Rasmussen for two-bridge knots). Applications include new restrictions on the Alexander polynomial of alternating knots.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/abdd3010317a58f7ffbb5b1cda9fbad24a612231

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Citations in the corpus

Grid Homology for Knots and Links (2015, methodology): “For more on doubly-pointed Heegaard diagrams, see [169, 172].”
A combinatorial description of knot Floer homology (2006, background): “in this case, the homology groups of the chain complex are trivial [18].”
A concordance invariant from the Floer homology of double branched covers (2005, background): “According to [25], alternating knots are perfect in the sense of [32, Definition 6.”
NOTIONS OF POSITIVITY AND THE OZSVÁTH–SZABÓ CONCORDANCE INVARIANT (2005, background): “Ozsváth and Szabó prove in [23] that τ(K) = σ(K) 2 for any knot which admits an alternating projection.”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “These diagrams were used in [OS03a] to prove Theorem 2.1.”
Holomorphic disks, link invariants and the multi-variable Alexander polynomial (2008, background): “In [18] we showed that the knot Floer homology of an alternating knot is determined explicitly in terms of the Alexander polynomial and signature of the knot.”
The decategorification of sutured Floer homology (2009, background): “The main theorem of [25] implies that ĤFK(L, g(R) + k − 1) is torsion-free and supported in a single homological grading, so the result follows from Corollary 6.”
Moving basepoints and the induced automorphisms of link Floer homology (2011, background): “All quasialternating knots are –thin [10; 5] and, out of the 85 prime knots up to nine crossings, 83 are quasi-alternating.”
Heegaard Floer homologies and contact structures (2002, background): “sequence of the calculation of CFK∞ for the trefoil given in Proposition 7.2 of [14], see also Subsection 2.3 above. Remark 3.3. A generalization of Lemma 3.2 to alternating, ﬁbered knots is given in [12] (see especially Corollary 1.7 of [12]). HEEGAARD FLOER HOMOLOGIES AND CONTACT STRUCTURES 17 4. ”
Computations of Heegaard-Floer knot homology (2006, background): “that all alternating knots K have α(K) = c(K) +2), which is fortunate since HFK\ is determined by more easily computed invariants (Alexander polynomial and signature) in the case of alternating knots [3]. In [7], Ozsvath and Szabo´ introduced a concordance invariant τ for knots in S3 which is deﬁned”
Khovanov homology and the slice genus (2004, result): “Combining the results of [4], [11], [13], [14], and [15] with some unpublished computations of the author, it appears that the value of τ has been determined for all but two knots of 10 crossings and fewer.”
On the geography and botany of knot Floer homology (2014, background): “For instance, the Floer homology of an alternating knot is determined by its Alexander polynomial and signature [66] and one can easily produce distinct alternating knots sharing these invariants.”