Geometry & Topology, Vol. 7 (2003) Paper no. 17, pages 615--639.

Knot Floer homology and the four-ball genus

Peter Ozsvath and Zoltan Szabo

Abstract. We use the knot filtration on the Heegaard Floer complex to define an integer invariant tau(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to Z. As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, tau gives sharp bounds on the four-ball genera of torus knots. As another illustration, we calculate the invariant for several ten-crossing knots.

Keywords. Floer homology, knot concordance, signature, 4-ball genus

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

DOI: 10.2140/gt.2003.7.615

E-print: arXiv:math.GT/0301149

Submitted to GT on 16 January 2003. (Revised 17 October 2003.) Paper accepted 21 September 2003. Paper published 22 October 2003.

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Peter Ozsvath and Zoltan Szabo
Department of Mathematics, Columbia University
New York 10025, USA
and
Department of Mathematics, Princeton University
New Jersey 08540, USA
Email: petero@math.columbia.edu, szabo@math.princeton.edu

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