On finite Thurston type orderings of braid groups

TL;DR

This paper proves that finite Thurston-type orderings on braid groups induce well-ordered structures on positive braids, using a new combinatorial normal form that generalizes existing braid normal forms.

Contribution

It introduces a new normal form for positive braids, called the C-normal form, and demonstrates its use in analyzing Thurston-type orderings on braid groups.

Findings

Positive braids under finite Thurston-type orderings form well-ordered sets.

The order type of these sets is ω^{ω^{n-2}}.

The C-normal form generalizes previous braid normal forms.

Abstract

We prove that for any finite Thurston-type ordering $<_{T}$ on the braid group\ $B_{n}$, the restriction to the positive braid monoid $(B_{n}^{+},<_{T})$ is a\ well-ordered set of order type $\omega^{\omega^{n-2}}$. The proof uses a combi\ natorial description of the ordering $<_{T}$. Our combinatorial description is \ based on a new normal form for positive braids which we call the $\C$-normal fo\ rm. It can be seen as a generalization of Burckel's normal form and Dehornoy's \ $\Phi$-normal form (alternating normal form).