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Knot surgery and primeness

  • Francisco González Acuña (a1) and Hamish Short (a2)
Abstract

The aim of this paper is to prove some new results towards answering the question: When does Dehn surgery on a knot give a non-prime manifold? This question has been raised on several occasions (see for instance [5] or [4]; concerning the latter see below). Recall that a 3-manifold is prime if, in any connected sum decomposition

one of M1, M2 is S3. (For standard definitions of low-dimensional topology see [2] or [16].)

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[2]R. H. Crowell and R. H. Fox . An Introduction to Knot Theory (Springer-Verlag, 1977).

[4]R. Fintushel and R. J. Stern . Constructing lens spaces by surgery on knots. Math. Z. 175 (1980), 3351; Correction, Math. Z. 178 (1981), 143.

[5]C. McA. Gordon . Dehn surgery and satellite knots. Trans. Amer. Math. Soc., 275 (1983) 687708.

[6]C. McA. Gordon and R. Litherland . Incompressible surfaces in 3-manifolds. Topology Appl. 18 (1984), 121144.

[13]L. Moser . Elementary surgery along a torus knot. Pacific J. Math. 38 (1971), 737745.

[14]K. Murasugi . On a certain subgroup of the group of an alternating link. Amer. J. Math. 85 (1963), 544550.

[15]F. H. Norwood . Every two-generator knot is prime. Proc. A.M.S., 86 (1982), 143147.

[17]C. P. Rourke . Presentations of the trivial group. In Topology of Low-Dimensional Manifolds, Lecture Notes in Math. vol. 722 (Springer-Verlag, 1979), 134143.

[18]J. Simon . Roots and centralizers of peripheral elements in knot groups. Math. Ann. 222 (1976), 205209.

[20]H. Zieschang , Über die Nielsensche Kürzungsmethode in freien Produkten mit Amalgam. Invent. Math. 10 (1970), 437.

[21]O. Viro . Linkings, 2-sheeted branched coverings and braids. Math. USSR Sbornik 16 (1972), 223226.

[22]J. S. Birman , and H. Hilden . The homeomorphism problem for S3. Bull. Amer. Math. Soc. 79 (1973), 10061010.

[23]J. P. Neuzil . Elementary surgery manifolds and the elementary ideals. Proc. Amer. Math. Soc. 68 (1978), 225228.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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