Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-th
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int