The Relaxation Normal Form of Braids is Regular

TL;DR

This paper proves that the relaxation normal form of braids, derived from tight laminations, is regular and prefix-closed, and provides a deterministic automaton for recognizing it.

Contribution

It establishes the regularity and prefix-closure of the relaxation normal form of braids and constructs a deterministic automaton for recognition.

Findings

The relaxation normal form is regular.

The normal form is prefix-closed.

A deterministic automaton recognizing the normal form is constructed.

Abstract

Braids can be represented geometrically as laminations of punctured disks. The geometric complexity of a braid is the minimal complexity of a lamination that represents it, and tight laminations are representatives of minimal complexity. These laminations give rise to a normal form of braids, via a relaxation algorithm. We study here this relaxation algorithm and the associated normal form. We prove that this normal form is regular and prefix-closed. We provide an effective construction of a deterministic automaton that recognizes this normal form.