The braid group $B_3$ in the framework of continued fractions

TL;DR

This paper leverages continued fractions to analyze the braid group B_3, providing efficient algorithms for solving the word and conjugacy problems with practical implementation advantages.

Contribution

It introduces simple, natural, and linear-time algorithms for key problems in B_3, based on its interpretation as a central extension of the modular group.

Findings

Linear-time algorithms for word and conjugacy problems in B_3

Algorithms are easy to implement and most efficient in literature

New insights into B_3 via continued fractions

Abstract

We use the classical interpretation of the braid group $B_3$ as a central extension of the modular group $\text{PSL}_2\left(\mathbb{Z}\right)$ to establish new and fundamental properties of $B_3$ using the theory of continued fractions. In particular, we give simple and natural linear time algorithms to solve the word and conjugacy problems in $B_3$. The algorithms introduced in this paper are easy to implement and are the most efficient algorithms in the literature to solve these problems in the braid group $B_3$.