An algorithm for the word problem in braid groups

TL;DR

This paper introduces a new algorithm for solving the word problem in braid groups, providing a finite-time solution and proving quadratic-time complexity conjecture in the 3-string case, with numerical evidence supporting its efficiency.

Contribution

The paper presents a novel algorithm for canonical braid representatives and proves its correctness and efficiency in specific cases, advancing computational methods in braid group theory.

Findings

Algorithm terminates in finite time

Proven to find minimal-length representatives for 3-string braids

Numerical evidence suggests quadratic-time complexity

Abstract

We suggest a new algorithm for finding a canonical representative of a given braid, and also for the harder problem of finding a $\sigma_1$-consistent representative. We conjecture that the algorithm is quadratic-time. We present numerical evidence for this conjecture, and prove two results: (1) The algorithm terminates in finite time. (2) The conjecture holds in the special case of 3-string braids - in fact, we prove that the algorithm finds a minimal-lenght representative for any 3-string braid.