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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].)

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[1]Conway, J. H.. An enumeration of knots and links, and some of their algebraic properties. In Computational Problems in Abstract Algebra (Pergamon Press, 1970), 329358.
[2]Crowell, R. H. and Fox, R. H.. An Introduction to Knot Theory (Springer-Verlag, 1977).
[3]Fenn, R.. Techniques of Geometric Topology. L.M.S. Lecture notes series, no. 57 (Cambridge University Press, 1983).
[4]Fintushel, R. and Stern, R. J.. Constructing lens spaces by surgery on knots. Math. Z. 175 (1980), 3351; Correction, Math. Z. 178 (1981), 143.
[5]Gordon, C. McA.. Dehn surgery and satellite knots. Trans. Amer. Math. Soc., 275 (1983) 687708.
[6]Gordon, C. McA. and Litherland, R.. Incompressible surfaces in 3-manifolds. Topology Appl. 18 (1984), 121144.
[7]Hatcher, A. and Thurston, W.. Incompressible surfaces in 2-bridge knot complements. Preprint.
[8]Kirby, R. C.. Problems in low dimensional manifold theory. Proc. Sympos. Pure Maths. 32 (A.M.S., 1978), 273312.
[9]Kim, P. K. and Tollefson, J. L.. Splitting the PL involutions of non-prime 3-manifolds. Mich. Math. J. 27 (1980), 259274.
[10]Lyndon, R. C. and Schupp, P.. Combinatorial Group Theory (Springer-Verlag, 1977).
[11]Montesinos, J. M.. Surgety on Links and Double Branched Covers. Annals of Math. Studies no. 84 (Princeton University Press, 1975).
[12]Magnus, W., Karass, A. and Solitar, D.. Combinatorial Group Theory (Dover, 1976).
[13]Moser, L.. Elementary surgery along a torus knot. Pacific J. Math. 38 (1971), 737745.
[14]Murasugi, K.. On a certain subgroup of the group of an alternating link. Amer. J. Math. 85 (1963), 544550.
[15]Norwood, F. H.. Every two-generator knot is prime. Proc. A.M.S., 86 (1982), 143147.
[16]Rolfsen, D.. Knots and, Links (Publish or Perish Inc., 1976).
[17]Rourke, C. P.. Presentations of the trivial group. In Topology of Low-Dimensional Manifolds, Lecture Notes in Math. vol. 722 (Springer-Verlag, 1979), 134143.
[18]Simon, J.. Roots and centralizers of peripheral elements in knot groups. Math. Ann. 222 (1976), 205209.
[19]Short, H. B.. Topological Methods in Group Theory. Thesis, University of Warwick, 1983.
[20]Zieschang, H., Über die Nielsensche Kürzungsmethode in freien Produkten mit Amalgam. Invent. Math. 10 (1970), 437.
[21]Viro, O.. Linkings, 2-sheeted branched coverings and braids. Math. USSR Sbornik 16 (1972), 223226.
[22]Birman, J. S., and Hilden, H.. The homeomorphism problem for S 3. Bull. Amer. Math. Soc. 79 (1973), 10061010.
[23]Neuzil, J. P.. 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
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