# 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.

## Key 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.