Geometry & Topology

Heegaard Floer homology and alternating knots

Peter Ozsváth and Zoltán Szabó

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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.

Article information

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

Received: 1 November 2002
Revised: 19 March 2003
Accepted: 20 March 2003
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R58: Floer homology
Secondary: 57M27: Invariants of knots and 3-manifolds 53D40: Floer homology and cohomology, symplectic aspects 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

alternating knots Kauffman states Floer homology


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.

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