Skip to main content
    • Aa
    • Aa

Knot surgery and primeness

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

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].)

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *