Distinguishing the generalized knot groups of square and granny knot analogues

Abstract

Given a knot [Formula: see text], we may construct a group [Formula: see text] from the fundamental group of [Formula: see text] by adjoining an [Formula: see text]th root of the meridian that commutes with the corresponding longitude. For [Formula: see text] these “generalized knot groups” determine [Formula: see text] up to reflection. The second author has shown that for [Formula: see text], the generalized knot groups of the square and granny knots can be distinguished by counting homomorphisms into a suitably chosen finite group. We extend this result to certain generalized knot groups of square and granny knot analogues [Formula: see text], [Formula: see text], constructed as connected sums of [Formula: see text]-torus knots of opposite or identical chiralities. More precisely, for coprime [Formula: see text] and [Formula: see text] satisfying a coprimality condition with [Formula: see text] and [Formula: see text], we construct an explicit finite group [Formula: see text] (depending on [Formula: see text], [Formula: see text] and [Formula: see text]) such that [Formula: see text] and [Formula: see text] can be distinguished by counting homomorphisms into [Formula: see text]. The coprimality condition includes all [Formula: see text] coprime to [Formula: see text]. The result shows that the difference between these two groups can be detected using a finite group.