Presentations of braid groups on graphs

Farley and Sabalka have given a method for computing presentations for braid groups on graphs by studying the corresponding configuration spaces of points on those graphs. In particular, this method relies on discrete Morse theory and on the fact that any graph must be sufficiently subdivided before applying it. We present a new and easier method for computing presentations for braid groups on graphs as follows. The idea is to define a kind of normalized configuration space and a cubical complex associated to it such that there is a homotopy equivalence between the fundamental group of the whole configuration space on the graph and the fundamental group of the cubical complex on the same graph. In this way, we are able to compute presentations for the braid groups on graphs by looking at the 2-skeleton of this cubical complex instead of studying the entire configuration space of n points on the graph.