Planar graphs and presentations of braid groups
The author observes that corresponding to a connected planar graph \(G\) with \(n\) vertices, there is a presentation of the braid group \(B_ n\) in which there is a generator for each edge of \(G\). The generators associated to non-adjacent edges commute, and there are also relations that arise from the vertices and faces of \(G\). The usual presentation of \(B_ n\) corresponds in this way to a simple path with \(n\) vertices, and in general if \(T\) is a spanning tree of \(G\) then the generators associated to edges of \(G\) not in \(T\) may be eliminated using simple Tietze transformations. Although these presentations of \(B_ n\) may be more complicated than the standard one, they can serve to highlight certain properties of \(B_ n\); in particular, symmetries of \(G\) may be associated with symmetries of \(B_ n\) which are not readily apparent in the usual presentation.
scientific article; zbMATH DE number 567455