Three-manifold mutations detected by Heegaard Floer homology

Abstract

Given an orientation-preserving self-diffeomorphism $\phi$ of a closed, orientable surface $S$ with genus at least two and an embedding $f$ of $S$ into a three-manifold $M$, we construct a mutant manifold by cutting $M$ along $f\left(S\right)$ and regluing by $f\phi f^{-1}$. We will consider whether there exist nontrivial gluings such that for any embedding, the manifold $M$ and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if $\phi$ is not isotopic to the identity map, then there exists an embedding of $S$ into a three-manifold $M$ such that the rank of the nontorsion summands of $\widehat{HF}$ 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 $\widehat{HF}$ of $M$ differs from that of its mutant.