Every braid admits a short sigma-definite representative

TL;DR

This paper proves that every braid has a sigma-definite representative that is also quasi-geodesic, confirming a longstanding conjecture and advancing the understanding of braid normal forms.

Contribution

It introduces a new normal form called the rotating normal form and proves that all braids have a quasi-geodesic sigma-definite representative, enhancing Dehornoy's classical result.

Findings

Every braid admits a sigma-definite quasi-geodesic representative

Introduces the rotating normal form for braids

Confirms a longstanding conjecture in braid theory

Abstract

A result by Dehornoy (1992) says that every nontrivial braid admits a sigma-definite word representative, defined as a braid word in which the generator sigma_i with maximal index i appears with exponents that are all positive, or all negative. This is the ground result for ordering braids. In this paper, we enhance this result and prove that every braid admits a sigma-definite word representative that, in addition, is quasi-geodesic. This establishes a longstanding conjecture. Our proof uses the dual braid monoid and a new normal form called the rotating normal form.