Describing elements of the genus-2 Goeritz group of $S^3$

TL;DR

This paper provides a finite generating set for the genus-2 Goeritz group of the 3-sphere, along with an algorithm to uniquely express its elements as words in this set, based on Dehn twists and a complexity measure.

Contribution

It introduces a finite generating set for the genus-2 Goeritz group of S^3 and an algorithm for unique word representation of its elements.

Findings

Finite generating set G_2 for the genus-2 Goeritz group of S^3.

An algorithm to express elements as words in G_2.

Uniqueness of the word description based on a complexity measure.

Abstract

In this article we present a finite generating set $G_2$ of $\mathcal{H}_2$, the genus-2 Goeritz group of $S^3$, in terms of Dehn twists about certain simple closed curves on the standard Heegaard surface. We present an algorithm that describes an element $\psi\in\mathcal{H}_2$ as a word in the alphabet of $G_2$ in a certain format. Using a complexity measure defined on reducing spheres, we show that such a description of $\psi$ is unique.