Peripherally specified homomorphs of knot groups

Abstract

Let G G be a group and let μ \mu and λ \lambda be elements of G G . Necessary and sufficient conditions are presented for the solution of the following problem: Is there a knot K K in S 3 {S^3} and a representation ρ : π 1 ( S 3 − K ) → G \rho :{\pi _1}({S^3} - K) \to G such that ρ ( m ) = μ \rho (m) = \mu and ρ ( l ) = λ \rho (l) = \lambda , where m m and l l are the meridian and longitude of K K ?