Generalised Knot Groups Distinguish the Square and Granny Knots

Given a knot K we may construct a group Gn(K) from the fundamental group of K by adjoining an nth root of the meridian that commutes with the corresponding longitude. These “generalised knot groups” were introduced independently by Wada and Kelly, and contain the fundamental group as a subgroup. The square knot SK and the granny knot GK are a well known example of a pair of distinct knots with isomorphic fundamental groups. We show that Gn(SK) and Gn(GK) are non-isomorphic for all n ≥ 2. This confirms a conjecture of Lin and Nelson, and shows that the isomorphism type of Gn(K), n ≥ 2, carries more information about K than the isomorphism type of the fundamental group. The appendix contains some results on representations of the trefoil group in PSL(2, p) that are needed for the proof.