Volume 7, issue 1 (2003)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Heegaard Floer homology and alternating knots

Peter Ozsváth and Zoltán Szabó

Geometry & Topology 7 (2003) 225–254

arXiv: math.GT/0209149

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
Mathematical Subject Classification 2000
Primary: 57R58
Secondary: 57M27, 53D40, 57M25
References
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Publication
Received: 1 November 2002
Revised: 19 March 2003
Accepted: 20 March 2003
Published: 24 March 2003
Proposed: John Morgan
Seconded: Yasha Eliashberg, Robion Kirby
Authors
Peter Ozsváth
Department of Mathematics
Columbia University
New York 10027
USA
Zoltán Szabó
Department of Mathematics
Princeton University
New Jersey 08540
USA