Genus two Goeritz equivalence in S3

Abstract

The Goeritz group of a genus g Heegaard splitting of a 3-manifold is the group of isotopy classes of orientation-preserving automorphisms of the manifold that preserve the Heegaard splitting. In the context of the standard genus 2 Heegaard splitting of S3, we introduce the concept of Goeritz equivalence of curves, present two algebraic obstructions to Goeritz equivalence of simple closed curves that are straightforward to compute, and provide families of examples demonstrating how these obstructions may be used.