On minimal crossing number braid diagrams and homogeneous braids

TL;DR

This paper investigates minimal crossing braid diagrams, establishing conditions for their minimality in homogeneous braids, and explores algebraic properties of braid monoids to derive growth rate bounds.

Contribution

It proves that homogeneous braid diagrams are minimal if and only if they are homogeneous and links braid monoids to Artin-Tits monoids, providing new algebraic insights.

Findings

Homogeneous braid diagrams are minimal iff they are homogeneous.

Monoids of alternating braids are right-angled Artin monoids.

A lower bound on the growth rate of braid groups is established.

Abstract

We study braid diagrams with a minimal number of crossings. Such braid diagrams correspond to geodesic words for the braid groups with standard Artin generators. We prove that a diagram of a homogeneous braid is minimal if and only if it is homogeneous. We conjecture that monoids of homogeneous braids are Artin-Tits monoids and prove that monoids of alternating braids are right-angled Artin monoids. Using this, we give a lower bound on the growth rate of the braid groups.