Factor Groups of Knot and LOT 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 $\mathbf{k}$ is a knot or virtual knot, one can study similar quotients $G(\mathbf{k},n)$ for the corresponding knot group. We identify a class of long virtual knots $\mathbf{k}$ for which $G(\mathbf{k},n)$ 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.