The proof of Birman’s conjecture on singular braid monoids

Paris, Luis

doi:10.2140/gt.2004.8.1281

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The proof of Birman's conjecture on singular braid monoids

Luis Paris

Geometry & Topology 8 (2004) 1281–1300

DOI: 10.2140/gt.2004.8.1281

arXiv: math.GR/0306422

Abstract

Let $B_{n}$ be the Artin braid group on $n$ strings with standard generators $σ_{1}, \dots, σ_{n - 1}$ , and let $S B_{n}$ be the singular braid monoid with generators $σ_{1}^{\pm 1}, \dots, σ_{n - 1}^{\pm 1}, τ_{1}, \dots, τ_{n - 1}$ . The desingularization map is the multiplicative homomorphism $η : S B_{n} \to ℤ [B_{n}]$ defined by $η (σ_{i}^{\pm 1}) = σ_{i}^{\pm 1}$ and $η (τ_{i}) = σ_{i} - σ_{i}^{- 1}$ , for $1 \leq i \leq n - 1$ . The purpose of the present paper is to prove Birman’s conjecture, namely, that the desingularization map $η$ is injective.

Keywords

singular braids, desingularization, Birman's conjecture

Mathematical Subject Classification 2000

Primary: 20F36

Secondary: 57M25. 57M27

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Publication

Received: 6 January 2004

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

Volume 8, issue 3 (2004)