Groebner-Shirshov basis for the braid group in the Birman-Ko-Lee-Garside generators

TL;DR

This paper develops Groebner-Shirshov bases for braid groups using Birman-Ko-Lee generators with a Garside element, providing a new algorithm for the word problem.

Contribution

It introduces a novel Groebner-Shirshov basis for braid groups in Birman-Ko-Lee generators with a Garside element, enabling new normal form and word problem algorithms.

Findings

New Groebner-Shirshov basis for braid groups

Algorithm for Birman-Ko-Lee Normal Form

Solution to the word problem in braid groups

Abstract

In this paper, we obtain Groebner-Shirshov (non-commutative Gr\"obner) bases for the braid groups in the Birman-Ko-Lee generators enriched by new ``Garside word" $\delta$. It gives a new algorithm for getting the Birman-Ko-Lee Normal Form in the braid groups, and thus a new algorithm for solving the word problem in these groups.