Heegaard Floer homology and alternating knots

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

Forward citations

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

Volume 7, issue 1 (2003)

Peter Ozsváth and Zoltán Szabó