A note on Artin-Markov normal form theorem for braid groups

TL;DR

This paper verifies the completeness of the Artin-Markov normal form theorem for braid groups by checking all compositions in the representation using Artin-Burau relations, confirming the validity of the approach.

Contribution

It provides a detailed verification of all compositions in the Artin-Markov normal form for braid groups, supporting previous claims of their completeness.

Findings

All compositions in the Artin-Markov presentation are checked and confirmed.

The representation of braid groups by Artin-Burau relations is validated.

The normal form theorem for braid groups is substantiated through this verification.

Abstract

In a recent paper by L. A. Bokut, V. V. Chaynikov and K. P. Shum in 2007, Braid group $B_n$ is represented by Artin-Burau's relations. For such a representation, it is told that all other compositions can be checked in the same way. In this note, we support this claim and check all compositions.