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Knot surgery and primeness

Published online by Cambridge University Press:  24 October 2008

Francisco González Acuña  and
Hamish Short
Francisco González Acuña
Affiliation:
Instituto de Matemáticas, Universidad National Autonoma de México
Hamish Short
Affiliation:
Department of Pure Mathematics, University of Liverpool
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Extract

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].)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 1 , January 1986 , pp. 89 - 102
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[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), 329–358.Google Scholar
[2]Crowell, R. H. and Fox, R. H.. An Introduction to Knot Theory (Springer-Verlag, 1977).CrossRefGoogle Scholar
[3]Fenn, R.. Techniques of Geometric Topology. L.M.S. Lecture notes series, no. 57 (Cambridge University Press, 1983).Google Scholar
[4]Fintushel, R. and Stern, R. J.. Constructing lens spaces by surgery on knots. Math. Z. 175 (1980), 33–51; Correction, Math. Z. 178 (1981), 143.CrossRefGoogle Scholar
[5]Gordon, C. McA.. Dehn surgery and satellite knots. Trans. Amer. Math. Soc., 275 (1983) 687–708.CrossRefGoogle Scholar
[6]Gordon, C. McA. and Litherland, R.. Incompressible surfaces in 3-manifolds. Topology Appl. 18 (1984), 121–144.CrossRefGoogle Scholar
[7]Hatcher, A. and Thurston, W.. Incompressible surfaces in 2-bridge knot complements. Preprint.Google Scholar
[8]Kirby, R. C.. Problems in low dimensional manifold theory. Proc. Sympos. Pure Maths. 32 (A.M.S., 1978), 273–312.Google Scholar
[9]Kim, P. K. and Tollefson, J. L.. Splitting the PL involutions of non-prime 3-manifolds. Mich. Math. J. 27 (1980), 259–274.CrossRefGoogle Scholar
[10]Lyndon, R. C. and Schupp, P.. Combinatorial Group Theory (Springer-Verlag, 1977).Google Scholar
[11]Montesinos, J. M.. Surgety on Links and Double Branched Covers. Annals of Math. Studies no. 84 (Princeton University Press, 1975).Google Scholar
[12]Magnus, W., Karass, A. and Solitar, D.. Combinatorial Group Theory (Dover, 1976).Google Scholar
[13]Moser, L.. Elementary surgery along a torus knot. Pacific J. Math. 38 (1971), 737–745.CrossRefGoogle Scholar
[14]Murasugi, K.. On a certain subgroup of the group of an alternating link. Amer. J. Math. 85 (1963), 544–550.CrossRefGoogle Scholar
[15]Norwood, F. H.. Every two-generator knot is prime. Proc. A.M.S., 86 (1982), 143–147.CrossRefGoogle Scholar
[16]Rolfsen, D.. Knots and, Links (Publish or Perish Inc., 1976).Google Scholar
[17]Rourke, C. P.. Presentations of the trivial group. In Topology of Low-Dimensional Manifolds, Lecture Notes in Math. vol. 722 (Springer-Verlag, 1979), 134–143.CrossRefGoogle Scholar
[18]Simon, J.. Roots and centralizers of peripheral elements in knot groups. Math. Ann. 222 (1976), 205–209.CrossRefGoogle Scholar
[19]Short, H. B.. Topological Methods in Group Theory. Thesis, University of Warwick, 1983.Google Scholar
[20]Zieschang, H., Über die Nielsensche Kürzungsmethode in freien Produkten mit Amalgam. Invent. Math. 10 (1970), 4–37.CrossRefGoogle Scholar
[21]Viro, O.. Linkings, 2-sheeted branched coverings and braids. Math. USSR Sbornik 16 (1972), 223–226.CrossRefGoogle Scholar
[22]Birman, J. S., and Hilden, H.. The homeomorphism problem for S 3. Bull. Amer. Math. Soc. 79 (1973), 1006–1010.CrossRefGoogle Scholar
[23]Neuzil, J. P.. Elementary surgery manifolds and the elementary ideals. Proc. Amer. Math. Soc. 68 (1978), 225–228.CrossRefGoogle Scholar