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

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.