On Generalized Knot Groups

Abstract

Generalized knot groups G_n(K) were introduced first by Wada and Kelly independently. The classical knot group is the first one G_1(K) in this series of finitely presented groups. For each natural number n, G_1(K) is a subgroup of G_n(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 $G_1(SK)\cong G_1(GK)$. From the presentations of G_n(SK) and G_n(GK), for n>1, it seems unlikely that G_n(SK) and G_n(GK) would be isomorphic to each other. We are able to show that for many finite groups H, the numbers of homomorphisms from G_n(SK) and G_n(GK) to H, respectively, are the same. Moreover, the numbers of conjugacy classes of homomorphisms from G_n(SK) and G_n(GK) to H, respectively, are also the same. It remains a challenge to us to show, as we would like to conjecture, that G_n(SK) and G_n(GK) are not isomorphic to each other for all n>1.