Mathematics – Geometric Topology
Scientific paper
2002-09-12
Geom. Topol. 7(2003) 225-254
Mathematics
Geometric Topology
Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper6.abs.html
Scientific paper
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.
Ozsvath Peter
Szabo Zoltan
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