The Relaxation Normal Form of Braids is Regular

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.