A simple presentation of the handlebody group of genus 2

TL;DR

This paper simplifies Wajnryb's complex presentation of the handlebody group of genus 2, making it more accessible and easier to understand by refining the existing algorithmic approach.

Contribution

It provides a simplified and clearer presentation of the handlebody group of genus 2, improving upon Wajnryb's original complex formulation.

Findings

Simplified the handlebody group presentation for genus 2

Reduced complexity of the original Wajnryb presentation

Enhanced understanding of the handlebody group's structure

Abstract

Using a similar algorithm to Hatcher-Thurston's algorithm for finding a presentation of the mapping class group of a surface, Wajnryb succeeded to find a presentation for the handlebody group. This is long and complicated. In this note I simplify Wajnryb's presentation for the handlebody group of genus g = 2.