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


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


[1] Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237 – 300. · Zbl 0633.57006 · doi:10.2307/1971311
[2] Jean Cerf, Sur les difféomorphismes de la sphère de dimension trois (\Gamma \(_{4}\)=0), Lecture Notes in Mathematics, No. 53, Springer-Verlag, Berlin-New York, 1968 (French). · Zbl 0164.24502
[3] C. D. Feustel and Wilbur Whitten, Groups and complements of knots, Canad. J. Math. 30 (1978), no. 6, 1284 – 1295. · Zbl 0373.55003 · doi:10.4153/CJM-1978-105-0
[4] David Gabai, Foliations and the topology of 3-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461 – 478. David Gabai, Foliations and the topology of 3-manifolds. III, J. Differential Geom. 26 (1987), no. 3, 479 – 536. · Zbl 0627.57012
[5] Leon Glass, A combinatorial analog of the Poincaré index theorem, J. Combinatorial Theory Ser. B 15 (1973), 264 – 268. · Zbl 0264.05112
[6] Klaus Johannson, Équivalences d’homotopie des variétés de dimension 3, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 23, Ai, A1009 – A1010 (French). · Zbl 0313.57003
[7] R. A. Litherland, Surgery on knots in solid tori. II, J. London Math. Soc. (2) 22 (1980), no. 3, 559 – 569. · Zbl 0508.57002 · doi:10.1112/jlms/s2-22.3.559
[8] Heinrich Tietze, Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatsh. Math. Phys. 19 (1908), no. 1, 1 – 118 (German). · JFM 39.0720.05 · doi:10.1007/BF01736688
[9] Friedhelm Waldhausen, Recent results on sufficiently large 3-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 21 – 38. · Zbl 0391.57011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.