On some decompositions of the 3-strand Singular Braid Group

Abstract

Let $S{B}_n$ be the singular braid group generated by braid generators ${\sigma }_i$ and singular braid generators ${\tau }_i$, $1\le i\le n-1$. Let $S{T}_n$ denote the group that is the kernel of the homomorphism that maps, for each $i$, ${\sigma }_i$ to the cyclic permutation $(i,i+1)$ and ${\tau }_i$ to $1$. In this paper we investigate the group $S{T}_3$. We obtain a presentation for $S{T}_3$. We prove that $S{T}_3$ is isomorphic to the singular pure braid group $S{P}_3$ on $3$ strands. We also prove that the group $S{T}_3$ is semi-direct product of a subgroup $H$ and an infinite cyclic group, where the subgroup $H$ is an HNN-extension of ${\mathbb{Z}}^2\ast {\mathbb{Z}}^2$.

minor revision. Corrected typos