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Dehn surgery on knots. (English) Zbl 0571.57008

Let M be a compact, connected orientable, irreducible 3-manifold, \(\partial M\) a torus, and M(c) obtained from M by attaching a solid torus with meridian c, where c is a given simple closed curve on \(\partial M\). Theorem. If \(\pi_ 1(M(c))\) and \(\pi_ 1(M(d))\) are cyclic, M not Seifert fibred, then the minimal geometric intersection number \(\Delta\) (c,d)\(\leq 1\). There are at most three non-isotopic curves c such that \(\pi_ 1(M(c))\) is cyclic. There are applications to Dehn surgeries on (non-trivial) knots \(K\subset S^ 3\). (Denote by K(r), \(r\in {\mathbb{Q}}\), the result of such a surgery [D. Rolfsen, Knots and links (1976; Zbl 0339.55004)].)
Corollary 1. If K is a torus knot, \(\pi_ 1(K(r))\) cyclic, then \(r\in {\mathbb{Z}}\). There are at most two such integers r and they must be successive. Corollary 2. If \(\pi_ 1(K(r))=1\), then \(r=1\) or \(r=-1\); but if \(\pi_ 1(K(\pm 1))=1\), then \(\pi_ 1(K(\mp 1))\neq 1\). Corollary 3. There are at most two knot types with homeomorphic complements. Corollary 4. \(\pi_ 1(K(r))\) is not cyclic if \(r\neq 0\), K amphicheiral. (Amphicheiral knots have property P.) Corollary 5. Knot types with Arf invariant 1 are determined by their complement. Corollary 6. Prime knots with isomorphic groups have homeomorphic complements. A proof is sketched using especially results of M. Culler and P. B. Shalen [Ann. Math., II. Ser. 117, 109-146 (1983; Zbl 0529.57005)], W. Jaco [Proc. Am. Math. Soc. 92, 288-292 (1984; Zbl 0564.57009)], K. Johannson [”On surfaces in one-relator 3-manifolds”, preprint] and W. Whitten [”Knot complements and groups”, preprint].
Reviewer: G.Burde

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
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[1] Marc Culler and Peter B. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math. (2) 117 (1983), no. 1, 109 – 146. · Zbl 0529.57005 · doi:10.2307/2006973
[2] C. McA. Gordon and R. A. Litherland, Incompressible planar surfaces in 3-manifolds, Topology Appl. 18 (1984), no. 2-3, 121 – 144. · Zbl 0554.57010 · doi:10.1016/0166-8641(84)90005-1
[3] William Jaco, Adding a 2-handle to a 3-manifold: an application to property \?, Proc. Amer. Math. Soc. 92 (1984), no. 2, 288 – 292. · Zbl 0564.57009
[4] Klaus Johannson, Homotopy equivalences of 3-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. · Zbl 0412.57007
[5] Klaus Johannson, On surfaces in one-relator 3-manifolds, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 157 – 192.
[6] Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. · Zbl 0854.57002
[7] Wilbur Whitten, Knot complements and groups, Topology 26 (1987), no. 1, 41 – 44. · Zbl 0607.57004 · doi:10.1016/0040-9383(87)90019-X
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