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 or the genus 2 hyperelliptic involution, then there exists an embedding of S into a three manifold M such that the rank of the nontorsion summands of the Heegaard Floer homology of M differs from that of its mutant.