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GEOMETRIC PRESENTATIONS FOR THE PURE BRAID GROUP

DAN MARGALIT

Department of Mathematics, 503 Boston Ave, Tufts University, Medford, MA 02155, USA

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and

JON McCAMMOND

Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA 93106-3080, USA

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https://doi.org/10.1142/S0218216509006859Cited by:17 (Source: Crossref)

Abstract

We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is why we describe them as geometric presentations. Motivated by a presentation for the full braid group that we call the "rotation presentation", we introduce presentations for the pure braid group that we call the "twist presentation" and the "swing presentation". From the point of view of mapping class groups, the swing presentation can be interpreted as stating that the pure braid group is generated by a finite number of Dehn twists and that the only relations needed are the disjointness relation and the lantern relation.

Keywords:

AMSC: 20F36, 20F05

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