On virtual cabling and a structure of 4-strand virtual pure braid group

Abstract

This paper is dedicated to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group [Formula: see text]. We define simplicial group [Formula: see text] and its simplicial subgroup [Formula: see text] which is generated by [Formula: see text]. Consequently, we describe [Formula: see text] as HNN-extension and find presentation of [Formula: see text] and [Formula: see text]. As an application to classical braids, we find a new presentation of the Artin pure braid group [Formula: see text] in terms of the cabled generators.

Extensions of Yang-Baxter sets

“…Doubling of the pure virtual braid group. By using the same ideas as in the work [15,24] on the classical braids in [12] it was introduced a simplicial group on the pure virtual braid groups with VAP n = V P n+1 , the face homomorphism…”