Abstract: We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index . In order to have the necessity part, graphs are organized into small classes so that one of the homological or cohomological characteristics of right-angled Artin groups can be applied. Finally we show that a given graph is planar iff the first homology of its 2-braid group is torsion-free, and we leave the corresponding statement for -braid groups as a conjecture along with a few other conjectures about graphs whose braid groups of index are right-angled Artin groups.
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Jee Hyoun Kim Affiliation:
Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea
Email:
kimjeehyoun@kaist.ac.kr
Ki Hyoung Ko Affiliation:
Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea
Email:
knot@kaist.ac.kr
Hyo Won Park Affiliation:
Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea
Email:
H.W.Park@kaist.ac.kr