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