A simple algorithm for finding short sigma-definite representatives

TL;DR

This paper introduces a straightforward algorithm that computes short, quasi-geodesic sigma-definite representatives of braids, ensuring the generator with the highest index appears uniformly positive or negative.

Contribution

The paper presents a novel, simple algorithm for efficiently finding short sigma-definite braid representatives, improving upon existing methods.

Findings

Algorithm produces shorter braid representatives

Ensures generator with maximal index appears uniformly

Operates efficiently for various braid sizes

Abstract

We describe a new algorithm which for each braid returns a quasi-geodesic sigma-definite word representative, defined as a braid word in which the generator sigma_i with maximal index i appears either only positively or only negatively.