Volume 8, issue 3 (2004)

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The proof of Birman's conjecture on singular braid monoids

Luis Paris

Geometry & Topology 8 (2004) 1281–1300
 arXiv: math.GR/0306422
Abstract

Let ${B}_{n}$ be the Artin braid group on $n$ strings with standard generators ${\sigma }_{1},\dots ,{\sigma }_{n-1}$, and let $S{B}_{n}$ be the singular braid monoid with generators ${\sigma }_{1}^{±1},\dots ,{\sigma }_{n-1}^{±1},{\tau }_{1},\dots ,{\tau }_{n-1}$. The desingularization map is the multiplicative homomorphism $\eta :\phantom{\rule{0.3em}{0ex}}S{B}_{n}\to ℤ\left[{B}_{n}\right]$ defined by $\eta \left({\sigma }_{i}^{±1}\right)={\sigma }_{i}^{±1}$ and $\eta \left({\tau }_{i}\right)={\sigma }_{i}-{\sigma }_{i}^{-1}$, for $1\le i\le n-1$. The purpose of the present paper is to prove Birman’s conjecture, namely, that the desingularization map $\eta$ is injective.

Keywords
singular braids, desingularization, Birman's conjecture
Mathematical Subject Classification 2000
Primary: 20F36
Secondary: 57M25. 57M27
Publication
Revised: 21 September 2004
Accepted: 21 September 2004
Published: 28 September 2004
Proposed: Joan Birman
Seconded: Robion Kirby, Cameron Gordon
Authors
 Luis Paris Institut de Mathématiques de Bourgogne Université de Bourgogne UMR 5584 du CNRS BP 47870 21078 Dijon cedex France