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Knots are determined by their complements. (English) Zbl 0678.57005

As indicated by the title, the authors solve the important problem: Does the homeomorphism type of its complement determine the type of a knot? Their positive answer to this 80 year old question of Tietze also holds in the orientation preserving case; namely they show that knots with complements homeomorphic by an orientation preserving homeomorphism are isotopic. The result follows from their Theorem 2, significant by itself: Nontrivial Dehn surgery on a nontrivial knot never yields \(S^ 3\). As a Corollary it then follows that prime knots with isomorphic groups are equivalent. Still another major result is Theorem 3. If a 3-manifold obtained by Dehn surgery on a nontrivial knot is reducible then it has a lens space as a connected summand. Ideas of Gabai and Litherland are used in the proofs which are, generally speaking, geometric, combinatoric and graph theoretic.
Reviewer: L.P.Neuwirth

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
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References:

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