Algebraic structures among virtual singular braids (Q6611794)

The classification of knots and links is equivalent to certain algebraic properties of classical braids as \textit{A. A. Markov} [Tr. Mat. Inst. Steklova 16, 3--53 (1945; Zbl 0061.02507)] showed. Similarly, the study of virtual knots, singular knots, virtual singular knots, and welded knots is intimately linked to that of virtual braids, singular braids, virtual singular braids, respectively.\N\NThe paper under review is devoted to the study of the algebraic properties of virtual singular braids. In particular, the authors show that the virtual singular braid monoid on \(n\) strands embeds in a group \(\mathrm{VSG}_{n}\), which they call the virtual singular braid group on \(n\) strands. The group \(\mathrm{VSG}_{n}\) contains a normal subgroup \(\mathrm{VSPG}_{n}\) of virtual singular pure braids and they show that \(\mathrm{VSG}_{n}\) is a semi-direct product of \(\mathrm{VSPG}_{n}\) and the symmetric group \(S_{n}\). The authors also represent \(\mathrm{VSPG}_{n}\) as a semi-direct product of \(n-1\) subgroups and they use these results to exhibit a normal form of words in the virtual singular braid group.