On Generalized Knot Groups

Generalized knot groups Gn(K) were introduced first by Wada and Kelly independently. The classical knot group is the first one G1(K) in this series of finitely presented groups. For each natural number n, G1(K) is a subgroup of Gn(K) so the generalized knot groups can be thought of as extensions of the classical knot group. For the square knot SK and the granny knot GK, we have an isomorphism G1(SK) ∼= G1(GK). From the presentations of Gn(SK) and Gn(GK), for n > 1, it seems unlikely that Gn(SK) and Gn(GK) would be isomorphic to each other. We are able to show that for many finite groups H , the numbers of homomorphisms from Gn(SK) and Gn(GK) to H , respectively, are the same. Moreover, the numbers of conjugacy classes of homomorphisms from Gn(SK) and Gn(GK) to H , respectively, are also the same. It remains a challenge to us to show, as we would like to conjecture, that Gn(SK) and Gn(GK) are not isomorphic to each other for all n > 1.