Epimorphisms of knot groups onto free products

Let k be a knot in S3. There is an epimorphism from π1(S3−k) onto a free product of two nontrivial cyclic groups sending a meridian to an element of length two iff k has property Q (Topology of Manifolds, Markham, Chicago, IL, 1970, pp. 195–199) that is if there is a closed surface F in S3 containing k, such that k is imprimitive in H1(X) and in H1(Y) where X and Y are the closures of the components of S3−F. We give answers to questions of Simon (1970) about properties Q, Q∗ and Q∗∗. Epimorphisms from knot groups onto torus knot groups are also studied and some results on property P and surgery are included.