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Knot Floer homology and the four-ball genus

Peter Ozsváth and Zoltán Szabó

Geometry & Topology 7 (2003) 615–639

arXiv: math.GT/0301149

Abstract

We use the knot filtration on the Heegaard Floer complex CF̂ to define an integer invariant τ(K) for knots. Like the classical signature, this invariant gives a homomorphism from the knot concordance group to . As such, it gives lower bounds for the slice genus (and hence also the unknotting number) of a knot; but unlike the signature, τ 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
Mathematical Subject Classification 2000
Primary: 57R58
Secondary: 57M25, 57M27
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Publication
Received: 16 January 2003
Revised: 17 October 2003
Accepted: 21 September 2003
Published: 22 October 2003
Proposed: Robion Kirby
Seconded: Tomasz Mrowka, Cameron Gordon
Authors
Peter Ozsváth
Department of Mathematics
Columbia University
New York 10025
USA
Zoltán Szabó
Department of Mathematics
Princeton University
New Jersey 08540
USA