Knot Floer homology and the four-ball genus

We use the knot filtration on the Heegaard Floer complex $\hat{C F}$ 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.

Keywords

Floer homology, knot concordance, signature, 4–ball genus

Mathematical Subject Classification 2000

Primary: 57R58

Secondary: 57M25, 57M27

References

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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

Volume 7, issue 2 (2003)

Peter Ozsváth and Zoltán Szabó