Distinguishing the generalised knot groups of square and granny knot analogues

Abstract

Given a knot $K$ we may construct a group $G_n(K)$ from the fundamental group of $K$ by adjoining an $n$th root of the meridian that commutes with the corresponding longitude. For $n\geq2$ these "generalised knot groups" determine $K$ up to reflection (Nelson and Neumann, 2008; arXiv:0804.0807). The second author has shown that for $n\geq2$, the generalised knot groups of the square knot $SK$ and the granny knot $GK$ can be distinguished by counting homomorphisms into a suitably chosen finite group (arXiv:0706.1807). We extend this result to certain generalised knot groups of square and granny knot analogues $SK_{a,b}=T_{a,b}\# T_{-a,b}$, $GK_{a,b}=T_{a,b}\# T_{a,b}$, constructed as connect sums of $(a,b)$-torus knots of opposite or identical chiralities. More precisely, for coprime $a,b\geq2$ and $n$ satisfying a certain coprimality condition with $a$ and $b$, we construct an explicit finite group $G$ (depending on $a$, $b$ and $n$) such that $G_n(SK_{a,b})$ and $G_n(GK_{a},b)$ can be distinguished by counting homomorphisms into $G$. The coprimality condition includes all $n\geq2$ coprime to $ab$. The result shows that the difference between these two groups can be detected using a finite group.