GEOMETRIC PRESENTATIONS FOR THE PURE BRAID GROUP

By Dany Margalit, Jon McCammond. 2006

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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. Mo- tivated 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c89304798353d95e7459c815072daad16f76be16

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Monodromy substitutions and rational blowdowns (2010, background): “Monodromy relations All of our relations may be derived from the basic lantern relation (indeed, the planar mapping class group may be given a presentation in which the lantern is the only relation, aside from commutation relations: see [11]).”
Planar open books, monodromy factorizations and symplectic fillings (2009, methodology): “(Note: The conventions in [21] are opposite ours and those used by most 4–manifold or symplectic topologists; they use left-handed Dehn twists as the positive generators of the presentation.”
The braided Thompson's groups are of type F∞ (2012, background): “n can be realized by using only 9 non-clone strands. Again, the standard presentation ts the bill. Example A.3.15. In the situation of arcs, the swing presentation for PB n, described in Section 4 of [MM09], provides an explicit example of AFn 5 n being 1- generating for n6 and AFn 9 n being 2-gener”
Injections of Artin groups (2005, background): “where the curves are as shown in Figure 9. This intuitive relation (which is already an interesting relation in the mapping class group) is explained more formally in the remark below (see also [ 30 , 33]).”