The Word Problem is Solvable for 3-free Artin groups in Quadratic Time

TL;DR

This paper presents a quadratic-time algorithm for solving the word problem in 3-free Artin groups, providing an efficient computational method and a way to transform geodesic words without increasing length.

Contribution

It introduces the first explicit quadratic-time algorithm for the word problem in 3-free Artin groups and shows how to relate geodesic words via length-preserving relations.

Findings

Quadratic-time algorithm for the word problem in 3-free Artin groups

Method to transform geodesic words using length-preserving relations

Proof that the word problem is solvable efficiently in this class of groups

Abstract

We give a quadratic-time explicit and computable algorithm to solve the word problem for Artin groups that do not contain any relations of length 3. Furthermore, we prove that, given two geodesic words representing the same element, one can obtain one from the other by using a set of homogeneous relations that never increase the word length.