Planar graphs and presentations of braid groups
type: Item
instance of: scholarly article
title: Planar graphs and presentations of braid groups
zbMATH Open document ID: 0819.20040
DOI: 10.1007/BF02572418
author: Vlad Sergiescu
published in: Mathematische Zeitschrift
publication date: 1994-09-22T00:00:00Z
full work available at URL: https://eudml.org/doc/174582
review text: 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.
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 567455
zbMATH Keywords: braid groups
zbMATH Keywords: generators
zbMATH Keywords: relations
zbMATH Keywords: Tietze transformations
zbMATH Keywords: presentations
zbMATH Keywords: symmetries
zbMATH Keywords: connected planar graphs
reviewed by: L. Traldi
MaRDI profile type: Publication
cites work: Configuration Spaces.
cites work: An acyclic extension of the braid group
cites work: Q5650865
description: scientific article; zbMATH DE number 567455