Quasitoric representation of generalized braids

TL;DR

This paper introduces generalized braid theories, proves that every oriented normal generalized knot can be represented as a quasitoric braid closure, and shows that these braids form a subgroup within the generalized braid group.

Contribution

It defines generalized braid theories, establishes the quasitoric representation of normal generalized knots, and demonstrates the subgroup structure of quasitoric braids.

Findings

Every oriented normal generalized knot is a quasitoric braid closure

Quasitoric normal generalized braids form a subgroup of the generalized braid group

A generating set for pure generalized braid theories is computed

Abstract

In this paper, we define generalized braid theories in alignment with the language of Fenn and Bartholomew for knot theories, and compute a generating set for the pure generalized braid theories. Using this, we prove that every oriented normal generalized knot is the closure of a quasitoric normal generalized braid. Further, we prove that the set of quasitoric normal generalized braids forms a subgroup of normal generalized braid group.