Geometry & Topology, Vol. 7 (2003) Paper no. 6, pages 225--254.

Heegaard Floer homology and alternating knots

Peter Ozsvath, Zoltan Szabo

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.

Keywords. Alternating knots, Kauffman states, Floer homology

AMS subject classification. Primary: 57R58. Secondary: 57M27, 53D40, 57M25.

DOI: 10.2140/gt.2003.7.225

E-print: arXiv:math.GT/0209149

Submitted to GT on 1 November 2002. (Revised 19 March 2003.) Paper accepted 20 March 2003. Paper published 24 March 2003.

Notes on file formats

Peter Ozsvath, Zoltan Szabo
Department of Mathematics, Columbia University
New York 10027, USA
Department of Mathematics, Princeton University
New Jersey 08540, USA

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