## Geometry & Topology

### Knot Floer homology and the four-ball genus

#### 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 $ℤ$. 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.

#### Article information

Source
Geom. Topol., Volume 7, Number 2 (2003), 615-639.

Dates
Received: 16 January 2003
Revised: 17 October 2003
Accepted: 21 September 2003
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883317

Digital Object Identifier
doi:10.2140/gt.2003.7.615

Mathematical Reviews number (MathSciNet)
MR2026543

Zentralblatt MATH identifier
1037.57027

#### Citation

Ozsváth, Peter; Szabó, Zoltán. Knot Floer homology and the four-ball genus. Geom. Topol. 7 (2003), no. 2, 615--639. doi:10.2140/gt.2003.7.615. https://projecteuclid.org/euclid.gt/1513883317

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