On some decompositions of the 3-strand singular braid group

Abstract

Let SBn be the singular braid group generated by braid generators σi and singular braid generators τi, 1≤i≤n−1. Let STn denote the group that is the kernel of the homomorphism that maps, for each i, σi to the cyclic permutation (i,i+1) and τi to 1. In this paper we investigate the group ST3. We obtain a presentation for ST3. We prove that ST3 is isomorphic to the singular pure braid group SP3 on 3 strands. We also prove that the group ST3 is semi-direct product of a subgroup H and an infinite cyclic group, where the subgroup H is an HNN-extension of Z2⁎Z2.