Characteristics of graph braid groups

Abstract

We give formulae for the first homology of the n-braid group and the pure 2-braid group over an arbitrary finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the n-braid group over the graph is torsion-free and the conjectures about the first homology of the pure 2-braid groups over graphs in [9] can be verified. As additional corollaries of these formulae, the n-braid group over a planar graph and the pure 2-braid group over any graph have a presentation whose relators are words of commutators, and the 2-braid group and the pure 2-braid group over a planar graph have a presentation whose relators are commutators, the latter of which was a conjecture in [8]. Finally we propose a conjecture that extends the conjecture in [8].