Genus 2 Goeritz Equivalence in $S^3$

TL;DR

This paper explores the concept of Goeritz equivalence in genus 2 splittings of the 3-sphere, introducing algebraic obstructions and demonstrating their application through examples.

Contribution

It introduces the notion of Goeritz equivalence of curves in genus 2 splittings and provides computable algebraic obstructions to determine this equivalence.

Findings

Two algebraic obstructions to Goeritz equivalence are identified.

Obstructions are straightforward to compute.

Examples demonstrate the application of these obstructions.

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 $S^3$, 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.