Knot Floer homology and the four-ball genus

Abstract

We use the knot filtration on the Heegaard Floer complex $\widehat{CF}$ 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 $\mathbb{Z}$. 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.