Distinguishing the generalised knot groups of square and granny knot analogues

TL;DR

This paper extends previous work on distinguishing knot groups by constructing specific finite groups that can differentiate generalized knot groups of square and granny knot analogues, including certain connect sums of torus knots.

Contribution

The paper introduces explicit finite groups capable of distinguishing generalized knot groups of specific knot analogues, extending prior results to a broader class of connect sums of torus knots.

Findings

Finite groups constructed depend on parameters a, b, n.

Distinction achieved for coprime a, b and suitable n.

Method applies to connect sums of torus knots with certain conditions.

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.