The proof of Birman's conjecture on singular braid monoids

Details The proof of Birman's conjecture on singular braid monoids The proof of Birman's conjecture on singular braid monoids

2003-06-30

arxiv.org/abs/math/0306422v2

Geom. Topol. 8(2004) 1281-1300

Mathematics

Group Theory

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper35.abs.html

Scientific paper

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.

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