Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes

TL;DR

This paper investigates the structure of braid graphs in simply-laced, triangle-free Coxeter systems, proving they are partial cubes and identifying Fibonacci links with Fibonacci cube structures.

Contribution

It introduces a unique factorization of reduced expressions and shows braid graphs decompose into box products, establishing their partial cube nature.

Findings

Braid graphs are partial cubes in certain Coxeter systems.

Reduced expressions have a unique factorization into links.

Braid graphs for Fibonacci links are Fibonacci cubes.

Abstract

In this paper, we study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid graph into a box product of the braid graphs for each link factor. When the Coxeter graph has no three-cycles, we use the decomposition to prove that braid graphs are partial cubes, i.e., can be isometrically embedded into a hypercube. For a special class of links, called Fibonacci links, we prove that the corresponding braid graphs are Fibonacci cubes.