Characteristics of Graph Braid Groups

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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 Farber and Hanbury (arXiv:1005.2300 [math.AT]) 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 Farley and Sabalka (J. Pure Appl. Algebra, 2012) and so we propose a similar conjecture for higher braid indices.
Publisher
SPRINGER
Issue Date
2012-12
Language
English
Article Type
Article
Keywords

CONFIGURATION-SPACE; MORSE-THEORY; 2 PARTICLES; TOPOLOGY

Citation

DISCRETE & COMPUTATIONAL GEOMETRY, v.48, no.4, pp.915 - 963

ISSN
0179-5376
DOI
10.1007/s00454-012-9459-8
URI
http://hdl.handle.net/10203/103487
Appears in Collection
MA-Journal Papers(저널논문)
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