The rotating normal form of braids is regular
This work addresses a theoretical problem in braid group theory, providing a formal proof of regularity for specific normal forms, which is incremental as it builds on existing concepts like Dehornoy's ordering.
The paper tackled the problem of recognizing rotating words in Birman-Ko-Lee monoids by constructing a finite-state automaton for all n ≥ 2, proving that the rotating normal form is regular, which also implies the regularity of a σ-definite normal form on the braid group.
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 $σ$-definite normal form defined on the whole braid group.