#### Volume 7, issue 2 (2003)

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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 $\stackrel{̂}{CF}$ to define an integer invariant $\tau \left(K\right)$ 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
##### Mathematical Subject Classification 2000
Primary: 57R58
Secondary: 57M25, 57M27
##### Publication
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