Knot Floer homology, genus bounds, and mutation

In an earlier paper, we introduced a collection of graded Abelian groups $\text{HFKa}(Y,K)$ 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 $\text{HFKa}$ contains more information than the Alexander polynomial and the signature of these knots; and they also illustrate the fact that $\text{HFKa}$ detects mutation. We also calculate $\text{HFKa}$ for certain pretzel knots, and knots with small crossing number ($n\le 9$). Our calculations prove that many of the knots considered here admit no Seifert fibered surgeries.