The rotating normal form of braids is regular

Abstract

Defined on Birman-Ko-Lee monoids, the rotating normal form has strong connections with the Dehornoy's braid ordering. It can be seen as a process for selecting between all the representative words of a Birman-Ko-Lee braid a particular one, called rotating word. In this paper we construct, for all n 2, a finite-state automaton which recognizes rotating words on n strands, proving that the rotating normal form is regular. As a consequence we obtain the regularity of a $\ddot{I}ƒ$-definite normal form defined on the whole braid group.

Erratum. The Lemma 4.1 of the previous version is incorrect, as pointed out by June Roupin. This lemma, is not used in the rest of the paper. We have replaced it with Definition 4.1