Centralisers in the braid group and singular braid monoid (Q1920465)

Let \(B_n\) be the Artin braid group with standard generators \(\sigma_1,\dots,\sigma_{n-1}\). \textit{J. Baez} [Lett. Math. Phys. 26, No. 1, 43-51 (1992; Zbl 0792.57002)] and \textit{J. Birman} [Bull. Am. Math. Soc., New Ser. 28, No. 2, 253-287 (1993; Zbl 0785.57001)] (independently) introduced the monoid \(SB_n\) of singular braids generated by, in addition to the \(\sigma_i\)'s, the elementary singular braids \(\tau_1,\dots,\tau_{n-1}\) each of which has a unique double point intersecting the \(i\)-th and \((i+1)\)-st strings respectively (\(1\leq i\leq n-1\)). The main result of this paper states coincidence of the sets \(\{x\in SB_n\mid x\sigma_j^r=\sigma^r_kx\}\) (\(r\neq 0\)), \(\{x\in SB_n\mid x\tau^r_j=\tau^r_kx\}\) (\(r\neq 0\)) for fixed \(j\), \(k\) by characterizing this set as those singular braids having certain proper/singular ribbons connecting \([j,j+1]\times\{0\}\) to \([k,k+1]\times\{1\}\). (Here, singular braids are topologically considered in \(\mathbb{C}\times[0,1]\) connecting \(\{1,\dots,n\}\times\{0,1\}\).) As an application, explicit generators of the centralizer of \(B_r\) (resp. \(SB_r\)) naturally embedded in \(B_n\) (resp. \(SB_n\)) (\(r<n\)) are described. J. Birman also conjectured injectivity of the natural monoid homomorphism \(\eta:SB_n\to\mathbb{Z}[B_n]\) (\(\sigma_i\mapsto\sigma_i\), \(\tau_i\mapsto\sigma_i-\sigma^{-1}_i\)). A proof for the injectivity on those at most two double points is given.