Homology Cobordisms of 3 Manifolds, Kn Ot Concordances, and Prime Kn Ots

Kirby and Lickorish [3] have previously obtained the second result using a different construction. This answers problem 13 in Gordon [2]. Results of Schubert [6] show that the knots constructed are prime. In the second half of this paper we will generalize his results to characterize prime "generalized" cable knots. If JSΓX is a knot in S x B and K2 is a knot in S , we can form the Kx cable of K2 by mapping S x B into S, with S x {0} going to K2. We will show that, with an obvious restriction, the Kt cable of K2 is prime if and only if Kx is prime in S 1 x B. The final section gives a large set of examples of prime knots in S x B. I would like to thank Rob Kirby for his helpful suggestions and conversations.