Knot Floer homology (introduced by Ozsvath-Szabo and independently by
Rasmussen) is a powerful tool for studying knots and links in the 3-sphere. In
particular, it gives rise to a numerical invariant, which provides a
nontrivial lower bound on the 4-dimensional genus of the knot. By deforming
the definition of knot Floer homology by a real number t from [0,2], we define
a family of homologies, and derive a family of numerical invariants with
similar properties. The resulting invariants provide a family of
homomorphisms on the concordance group. One of these homomorphisms can be
used to estimate the unoriented 4-dimensional genus of the knot. We will
review the basic constructions for knot Floer homology and the deformed
theories and discuss some of the applications. This is joint work with
P. Ozsvath and Z. Szabo.