Geometry & Topology

Heegaard Floer homology and alternating knots

Abstract

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.

Article information

Source
Geom. Topol., Volume 7, Number 1 (2003), 225-254.

Dates
Revised: 19 March 2003
Accepted: 20 March 2003
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883097

Digital Object Identifier
doi:10.2140/gt.2003.7.225

Mathematical Reviews number (MathSciNet)
MR1988285

Zentralblatt MATH identifier
1083.57013

Citation

Ozsváth, Peter; Szabó, Zoltán. Heegaard Floer homology and alternating knots. Geom. Topol. 7 (2003), no. 1, 225--254. doi:10.2140/gt.2003.7.225. https://projecteuclid.org/euclid.gt/1513883097

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