#### 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.

Notes on file formats
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

GT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**