Geometry & Topology, Vol. 8 (2004) Paper no. 35, pages 1281--1300.

The proof of Birman's conjecture on singular braid monoids

Luis Paris

Abstract. Let B_n be the Artin braid group on n strings with standard generators sigma_1, ..., sigma_{n-1}, and let SB_n be the singular braid monoid with generators sigma_1^{+-1}, ..., sigma_{n-1}^{+-1}, tau_1, ..., tau_{n-1}. The desingularization map is the multiplicative homomorphism eta: SB_n --> Z[B_n] defined by eta(sigma_i^{+-1}) = _i^{+-1} and eta(tau_i) = sigma_i - sigma_i^{-1}, for 1 <= i <= 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

AMS subject classification. Primary: 20F36. Secondary: 57M25. 57M27.

DOI: 10.2140/gt.2004.8.1281

E-print: arXiv:math.GR/0306422

Submitted to GT on 6 January 2004. (Revised 21 September 2004.) Paper accepted 21 September 2004. Paper published 28 September 2004.

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Luis Paris
Institut de Mathematiques de Bourgogne, Universite de Bourgogne
UMR 5584 du CNRS, BP 47870, 21078 Dijon cedex, France

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