Gr\"obner-Shirshov bases for braid groups in Adyan-Thurston generators

TL;DR

This paper develops a Gr"obner-Shirshov basis for braid groups using Adyan-Thurston generators, leading to new algorithms for normal forms and insights into the structure of braid semigroups.

Contribution

It introduces a novel Gr"obner-Shirshov basis for braid groups in Adyan-Thurston generators, providing new algorithms and proofs related to braid semigroups.

Findings

New Gr"obner-Shirshov basis for braid groups

Algorithm for Adyan-Thurston normal form

Proof that braid semigroup is a subsemigroup

Abstract

In this paper, we give a Gr\"obner-Shirshov basis of the braid group $B_{n+1}$ in Adyan-Thurston generators. We also deal with the braid group of type $\bf{B}_{n}$. As results, we obtain a new algorithm for getting the Adyan-Thurston normal form, and a new proof that the braid semigroup $B^+_{n+1}$ is the subsemigroup in $B_{n+1}$.