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The decision problems in the title are solved using the solutions of the word and conjugacy problems obtained by Lyndon and Schupp, respectively, for certain classes of groups. 1. Introduction. Each tame knot K has a knot group G whose word problem is solvable by Waldhausen's result (o). Here we give additional results when K is also prime, alternating, and nontrivial. We show that the free product of G and a free group on one generator Xo is a group 77 which falls into one of the categories for which Lyndon (4) and Schupp (5) solved the word and conjugacy problems, re- spectively. (77 has a presentation satisfying the properties C(4) and T3 (4, p. 219).) We thereby obtain another solution of the word problem for G and a solution of the conjugacy problem for G. In the proof, the given knot is replaced with an equivalent one having a knot-diagram with special properties (referred to as a common knot-diagram). Presentations of G and of another group 77 are obtained from a knot-diagram, using Dehn's method (3, p. 157). When the knot-diagram is common, the presentation of 77 (as a factor group of a free group 7) satisfies C(4) and T3. An automorphism of P yields a second presentation of H in which one generator Xo is not mentioned in the defining relations. Adding the relation x0= 1 to this second presentation gives a presentation of G. Thus H is a free product as claimed. 2. Common knot-diagrams. Let K be a nontrivial polygonal knot in 3-space P3. Let K he the range of a continuous function / defined on (O, l), with/(0) =/(l). Assume K is in regular position with re- spect to the projection p which sends (x, y, z) to (x, y, 0). (Knot- theoretic terminology is taken from Crowell and Fox (2).) The knot-

The word and conjugacy problems for the knot group of any tame, prime, alternating knot