On virtual singular braid groups

TL;DR

This paper explores the algebraic structure of the virtual singular braid group, introducing invariants, homomorphisms, and quotients, and providing explicit presentations especially for the case n=2.

Contribution

It systematically analyzes algebraic properties of the virtual singular braid group, including invariants, homomorphisms, and structural decompositions, extending understanding of these complex groups.

Findings

Defined numerical invariants from exponent sums in VSG_n

Described kernels of homomorphisms onto abelian groups

Provided explicit presentations for n=2 case

Abstract

The virtual singular braid group arises as a natural common generalization of classical singular braid groups and virtual braid groups. In this paper, we study several algebraic properties of the virtual singular braid group $VSG_n$. We introduce numerical invariants for virtual singular braids arising from exponent sums of words in $VSG_n$, and describe explicitly the kernels of the associated homomorphisms onto abelian groups. We then determine all group homomorphisms, up to conjugation, from $VSG_n$ to the symmetric group $S_n$, and obtain corresponding semi-direct product decompositions. In the particular case $n=2$, we provide explicit presentations and algebraic descriptions of the kernels. Moreover, we show that certain relations are forbidden in $VSG_n$, and we introduce and study natural quotients of the virtual singular braid group, including welded and unrestricted versions, for which analogous structural results are obtained.