Dana Ernst (Northern Arizona University)

In this talk, we will discuss the architecture of braid graphs in simply-laced Coxeter systems. It turns out 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 corresponding Coxeter graph is triangle-free, each braid graph is a median graph (i.e., for every triple of vertices, there is a unique vertex, called the median, that belongs to shortest paths between each pair). One consequence of this result is that every braid graph in simply-laced and triangle-free Coxeter systems is the one-skeleton of a CAT(0) cube complex. Moreover, every such graph can be isometrically embedded into a hypercube.