TITLE: Complete knot invariant?
QUESTION [26 upvotes]: I've seen a couple papers (that I now can't find) that say that in his paper "On irreducible 3-manifolds which are sufficiently large" Waldhausen proved that the data $\pi_1(\partial (S^3\setminus K)) \to \pi_1(S^3\setminus K)$ is a complete knot invariant. However, the word "knot" doesn't appear in this paper (although the phrase "to avoid an orgy of notation" does :-). Is the claimed result a straightforward corollary of his main results? Or am I looking at the wrong paper?

REPLY [3 votes]: This topic is treated in G. Burde & H. Zieschang's book, Knots, 2nd edition, p. 40.