The proof of Birman's conjecture on singular braid monoids

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 ${\sigma }_1^{\pm 1},\dots ,{\sigma }_{n-1}^{\pm 1},{\tau }_1,\dots ,{\tau }_{n-1}$. The desingularization map is the multiplicative homomorphism $\eta :S{B}_n\to \mathbb{Z}\left[B_n\right]$ defined by $\eta \left({\sigma }_i^{\pm 1}\right)={\sigma }_i^{\pm 1}$ 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.