Knot Floer homology, genus bounds, and mutation

In an earlier paper, we introduced a collection of graded Abelian groups HFK(Y,K) associated to knots in a three-manifold. The aim of the present paper is to investigate these groups for several specific families of knots, including the Kinoshita–Terasaka knots and their “Conway mutants”. These results show that HFK contains more information than the Alexander polynomial and the signature of these knots; and they also illustrate the fact that HFK detects mutation. We also calculate HFK for certain pretzel knots, and knots with small crossing number (n⩽9). Our calculations give obstructions to certain Seifert fibered surgeries on the knots considered here.