Markov and Artin normal form theorem for braid groups

TL;DR

This paper applies Groebner--Shirshov basis techniques to braid groups to derive the Artin--Markov normal form, simplifying the understanding of their algebraic structure.

Contribution

It introduces a new proof of the Artin--Markov normal form for braid groups using Groebner--Shirshov basis methods, providing a clearer algebraic framework.

Findings

Reobtained the Artin--Markov--Ivanovsky normal form

Demonstrated the effectiveness of Groebner--Shirshov basis in braid group theory

Simplified the derivation of normal forms for braid groups

Abstract

In this paper we will present the results of Artin--Markov on braid groups by using the Groebner--Shirshov basis. As a consequence we can reobtain the normal form of Artin--Markov--Ivanovsky as an easy corollary.