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Given a knot K we may construct a group G_n(K) from the fundamental group ofK by adjoining an nth root of the meridian that commutes with the correspondinglongitude. These "generalised knot groups" were introduced independently byWada and Kelly, and contain the fundamental group as a subgroup. The squareknot SK and the granny knot GK are a well known example of a pair of distinctknots with isomorphic fundamental groups. We show that G_n(SK) and G_n(GK) arenon-isomorphic for all n>1. This confirms a conjecture of Lin and Nelson, andshows that the isomorphism type of G_n(K), n>1, carries more information aboutK than the isomorphism type of the fundamental group. An appendix by DavidSavitt contains some results on representations of the trefoil group inPSL(2,p) that are needed for the proof.

Generalised knot groups distinguish the square and granny knots (with an appendix by David Savitt)