Centralisers in the braid group and singular braid monoid
type: Item
instance of: scholarly article
title: Centralisers in the braid group and singular braid monoid
zbMATH Open document ID: 0869.20024
author: Roger Fenn
author: J. Zhu
author: Dale Rolfsen
published in: L'Enseignement Mathématique. 2e Série
publication date: 1997-09-03T00:00:00Z
review text: Let \(B_n\) be the Artin braid group with standard generators \(\sigma_1,\dots,\sigma_{n-1}\). \textit{J. Baez} [Lett. Math. Phys. 26, No. 1, 43-51 (1992; Zbl 0792.57002)] and \textit{J. Birman} [Bull. Am. Math. Soc., New Ser. 28, No. 2, 253-287 (1993; Zbl 0785.57001)] (independently) introduced the monoid \(SB_n\) of singular braids generated by, in addition to the \(\sigma_i\)'s, the elementary singular braids \(\tau_1,\dots,\tau_{n-1}\) each of which has a unique double point intersecting the \(i\)-th and \((i+1)\)-st strings respectively (\(1\leq i\leq n-1\)). The main result of this paper states coincidence of the sets \(\{x\in SB_n\mid x\sigma_j^r=\sigma^r_kx\}\) (\(r\neq 0\)), \(\{x\in SB_n\mid x\tau^r_j=\tau^r_kx\}\) (\(r\neq 0\)) for fixed \(j\), \(k\) by characterizing this set as those singular braids having certain proper/singular ribbons connecting \([j,j+1]\times\{0\}\) to \([k,k+1]\times\{1\}\). (Here, singular braids are topologically considered in \(\mathbb{C}\times[0,1]\) connecting \(\{1,\dots,n\}\times\{0,1\}\).) As an application, explicit generators of the centralizer of \(B_r\) (resp. \(SB_r\)) naturally embedded in \(B_n\) (resp. \(SB_n\)) (\(r<n\)) are described. J. Birman also conjectured injectivity of the natural monoid homomorphism \(\eta:SB_n\to\mathbb{Z}[B_n]\) (\(\sigma_i\mapsto\sigma_i\), \(\tau_i\mapsto\sigma_i-\sigma^{-1}_i\)). A proof for the injectivity on those at most two double points is given.
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 915930
zbMATH Keywords: monoids of singular braids
zbMATH Keywords: Artin braid groups
zbMATH Keywords: generators
zbMATH Keywords: ribbons
zbMATH Keywords: centralizers
zbMATH Keywords: injectivity
reviewed by: Hiroaki Nakamura
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 915930