Characteristics of graph braid groups

We give formulae for the first homology of the $n$-braid group and the pure 2-braid group over a 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 \cite{FH} can be verified. We discover more characteristics of graph braid groups: 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 was a conjecture in \cite{FS2} and so we propose a similar conjecture for higher braid indices.

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