We define a concordance invariant, ε(K), associated to the knot Floer complex of K, and give a formula for the Ozsváth-Szabó concordance invariant τ of K p,q , the (p, q)-cable of a knot K, in terms of p, q, τ (K), and ε(K). We also describe the behavior of ε under cabling, allowing one to compute τ of iterated cables. Various properties and applications of ε are also discussed.

Based on work of Rasmussen [Ras03], we construct a concordance invariant associated to the knot Floer complex, and exhibit examples in which this invariant gives arbitrarily better bounds on the 4-ball genus than the Ozsváth-Szabó τ invariant.

In this survey article, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsváth and Szabó. Along the way, we prove that if two knots are concordant, then their knot Floer complexes satisfy a certain type of stable equivalence.

Abstract. We define a new smooth concordance homomorphism based on the knot Floer complex and an associated concordance invariant, ε. As an application, we show that an infinite family of topologically slice knots are independent in the smooth concordance group.