A User's Guide to the Mapping Class Group: Once Punctured Surfaces

TL;DR

This paper provides a practical, efficient algorithm for solving the word problem in the mapping class group of a once-punctured surface, enabling computations with a unique normal form.

Contribution

It introduces a quadratic time algorithm that computes a unique normal form for elements of the mapping class group using chord diagrams and elementary moves.

Findings

Quadratic time complexity for the word problem algorithm

Efficient implementation with pencil and paper

Unique normal form for group elements

Abstract

This document is a practical guide to computations using an automatic structure for the mapping class group of a once-punctured, oriented surface $S$. We describe a quadratic time algorithm for the word problem in this group, which can be implemented efficiently with pencil and paper. The input of the algorithm is a word, consisting of ``chord diagrams'' of ideal triangulations and elementary moves, which represents an element of the mapping class group. The output is a word called a ``normal form'' that uniquely represents the same group element.