Factor Groups of Knot and LOT Groups

TL;DR

This paper explores the properties of certain quotients of knot groups, identifying conditions under which these quotients are infinite, especially for long virtual knots with non-positively curved Wirtinger complexes.

Contribution

It extends Coxeter's classical results to virtual knot groups, identifying a class of long virtual knots with infinite quotients based on geometric properties.

Findings

Certain long virtual knots have infinite quotients G(k, n) for n ≥ 2.

Wirtinger complexes of these knots are non-positively curved squared complexes.

The study generalizes classical results from braid groups to virtual knot groups.

Abstract

A classical result of H. S. M. Coxeter asserts that a certain quotient $B(m,n)$ of the braid group $B(m)$ on $m$ strands is finite if and only if $(m,n)$ corresponds to the type of one of the five Platonic solids. If ${\bf k}$ is a knot or virtual knot, one can study similar quotients $G({\bf k}, n)$ for the corresponding knot group. We identify a class of long virtual knots ${\bf k}$ for which $G({\bf k}, n)$ is infinite for $n\ge 2$. The main feature of these long virtual knots is that their Wirtinger complexes are non-positively curved squared complexes.