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The group system of a tame knot [l] consists of the fundamental group of the complement of the knot, along with the conjugacy class of subgroups determined by the fundamental group of the boundary of a small tube around the knot. If the knot is non trivial, these subgroups are free abelian of rank 2, as was correctly pointed out, but incorrectly proved by M. Dehn in 1910 [2]. In fact this was the purpose of Dehn's Lemma, which was not properly proven until 1957, when Papakyriakopoulos gave a complete proof [3]. It has long been believed that the group system of a knot in S* determines the topological type of the complement. Here it will be shown that if the commutator subgroup of a knot group is finitely generated, then the group system determines the topological type of the complement. Let k denote a tame knot in Sz, and let M(k) denote the closure of the complement of a regular neighborhood of k. Then it is well known that the topological type of M(k) depends only on the type of k. (See for example [4].) Now suppose k\, k2, two tame knots in S3 are given, and there exists an isomorphism p, mapping iri(M(ki)) onto ■Ki(M(k2)) and mapping the conjugacy class of wi(M(ki)) onto that of ■wi(M(k2)). We may assume without loss of generality that the base points, xi, x2, for 7ri(Af(¿i)), iri(M(k2)) are located on the boundary in each case; also by a suitable automorphism of iri(M(k2)), p may be assumed to map iri(M(ki)) Ciri(M(ki)) onto tti(M(k2)) Ciri(M(k2)). Let «i, k, denote a meridian and longitude of M(ki), (by a meridian is meant a simple representative generating Hi(M(ki)), and homotopic to 0 in S3 — int(M(ki)), and by a longitude is meant a simple curve which bounds in M(ki), and with mi generates Ti(M(ki)), let m2, l2 denote a meridian and longitude of M(k2). Then

The algebraic determination of the topological type of the complement of a knot