Groebner-Shirshov basis for the braid group in the Artin-Garside generators

TL;DR

This paper establishes a Groebner-Shirshov basis for the braid group in Artin-Garside generators, leading to a new algorithm for Garside normal form and a proof of the braid semigroup's subsemigroup property.

Contribution

It introduces a Groebner-Shirshov basis for the braid group in Artin-Garside generators, enabling new computational methods and theoretical insights.

Findings

New algorithm for Garside normal form

Proof that the braid semigroup is a subsemigroup

Establishment of a Groebner-Shirshov basis for the braid group

Abstract

In this paper, we give a Groebner-Shirshov basis of the braid group $B_{n+1}$ in the Artin--Garside generators. As results, we obtain a new algorithm for getting the Garside normal form, and a new proof that the braid semigroup $B^+{n+1}$ is the subsemigroup in $B_{n+1}$.