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

TL;DR

This paper introduces a new generating set for the virtual pure braid group through cabling, describes its structure as an HNN-extension, and provides a new presentation for the classical pure braid group P4.

Contribution

It presents a novel cabling construction for virtual braids and offers a new algebraic presentation for P4 based on cabled generators.

Findings

New generating set for VP_n via cabling

VP_4 described as HNN-extension

New presentation of P_4 with cabled generators

Abstract

This article is dedicate to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group $VP_n$. Consequently we describe $VP_4$ as HNN-extension. As an application to classical braids, we find a new presentation of the Artin pure braid group $P_4$ in terms of the cabled generators.