Minimal generating sets and the abelianization for the quasitoric braid group

TL;DR

This paper introduces two minimal generating sets for the quasitoric braid group and determines its abelianization, providing insights into its algebraic structure and minimal generating elements.

Contribution

It presents the first explicit minimal generating sets for the quasitoric braid group and computes its abelianization, advancing understanding of its algebraic properties.

Findings

Two minimal generating sets for the quasitoric braid group identified

The abelianization of the quasitoric braid group explicitly determined

Minimality of generators established through lower bounds

Abstract

A toric braid is a braid whose closure is a torus link in $\mathbb{R}^3$. Manturov generalized toric braids that is called quasitoric braids and showed that the subset of quasitoric braids in the classical braid group is a subgroup of the braid group. We call this subgroup the quasitoric braid group. In this paper, we give two minimal generating sets for the quasitoric braid group and determine its abelianization. The minimalities of these two generating sets are obtained from a lower bound by the number of generators for the abelianization.