The rotating normal form of braids is regular

TL;DR

This paper proves that the rotating normal form of braids, defined on Birman-Ko-Lee monoids, is regular by constructing a finite-state automaton, which has implications for the regularity of related braid normal forms.

Contribution

It constructs a finite-state automaton recognizing rotating words for all n, establishing the regularity of the rotating normal form of braids.

Findings

Finite-state automaton for rotating words on n strands

Proof of regularity of the rotating normal form

Implication for the regularity of the $\sigma$-definite normal form

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 $\sigma$-definite normal form defined on the whole braid group.