  Given a self-diffeomorphism h of a closed, orientable surface S and an
embedding f of S into a three-manifold M, we construct a mutant manifold N by
cutting M along f(S) and regluing by h. We will consider whether there are any
gluings such that for any embedding, the manifold and its mutant have
isomorphic Heegaard Floer homology. In particular, we will demonstrate that if
the gluing is not isotopic to the identity, then there exists an embedding of S
into a three-manifold M such that the rank of the non-torsion summands of the
Heegaard Floer homology of M differs from that of its mutant. We will also show
that if the gluing map is isotopic to neither the identity nor the genus-two
hyperelliptic involution, then there exists an embedding of S into a
three-manifold M such that the total rank of the Heegaard Floer homology of M
differs from that of its mutant.
