Knot Floer homology and the four-ball genus

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.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper17.abs.htm

On cobordisms between knots, braid index, and the Upsilon-invariant

“…Here, τ denotes the Ozsváth-Szabó concordance invariant, whose value for positive torus knots equals the slice genus [OS03b].…”

A combinatorial description of knot Floer homology

“…We call attention to another other construction, which gives a concordance invariant τ (K) for knots [21], [27]. This is a homomorphism from the smooth concordance group of knots to the integers, which can be used to bound the four-ball genus of knots, giving an alternate proof of the theorem of Kronheimer and Mrowka [13] confirming Milnor's conjecture for the unknotting numbers of torus knots.…”

Khovanov homology, sutured Floer homology and annular links

“…Heegaard Floer homology as studied by Ozsváth and Szabó [19] and Khovanov homology [10] have transformed the landscape of low-dimensional topology in the past decade, generating a wealth of applications, most notably to questions in knot concordance (cf Ozsváth and Szabó [16] and Rasmussen [27; 26]), Dehn surgery (cf Ozsváth and Szabó [21] and Watson [29]) and contact geometry (cf Ozsváth and Szabó [20] and Plamenevskaya [25]). The philosophies underlying the theories' constructions are quite different, yet there are intriguing connections between the two.…”