  In an earlier paper, we introduced a knot invariant for a null-homologous
knot K in an oriented three-manifold Y, which is closely related to the
Heegaard Floer homology of Y. In this paper we investigate some properties of
these knot homology groups for knots in the three-sphere. We give a
combinatorial description for the generators of the chain complex and their
gradings. With the help of this description, we determine the knot homology for
alternating knots, showing that in this special case, it depends only on the
signature and the Alexander polynomial of the knot (generalizing a result of
Rasmussen for two-bridge knots). Applications include new restrictions on the
Alexander polynomial of alternating knots.
