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Only integral Dehn surgeries can yield reducible manifolds

Published online by Cambridge University Press:  24 October 2008

C. McA. Gordon  and
J. Luecke
C. McA. Gordon
Affiliation:
University of Texas, Austin, TX 78712, U.S.A.
J. Luecke
Affiliation:
Courant Institute, New York University, New York, NY 10012, U.S.A.
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Let K be a knot in S3, and let K(a/b) denote the closed oriented 3-manifold obtained by a/b-Dehn surgery on K. (We parametrize the Dehn surgeries on K by ℚ ∩ {1/0} as in [9]; in particular, K(1/0) = S3.) If K is a (p, q)-cable knot, then K(pq) is always reducible (see Section 3), and it is conjectured by González-Acuña and Short in [5] that these are the only examples where Dehn surgery on a knot in S3 yields a reducible manifold. One of the results in this direction proved in [5] is that if π1 (K(a/b)) is a non-trivial free product then |b| ≤ 5. We show that this may be sharpened to the assertion in the title.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 1 , July 1987 , pp. 97 - 101
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1] Baumslag, G., Morgan, J. and Shalen, P. B.. Generalized triangle groups. Math. Proc. Cambridge Philos. Soc. 102 (1987), 25–31.CrossRefGoogle Scholar
[2] Boyer, S.. On proper powers in free products and Dehn surgery. Preprint.Google Scholar
[3] Culler, M., Gordon, C. McA., Luecke, J. and Shalen, P. B.. Dehn surgery on knots. Annals of Math, (to appear).Google Scholar
[4] Gabai, D.. Foliations and the topology of 3-manifolds II. Preprint, MSRI, Berkeley, 1985.Google Scholar
[5] González-Acuña, F. and Short, H.. Knot surgery and primeness. Math. Proc. Cambridge Philos. Soc. 99 (1986), 89–102.CrossRefGoogle Scholar
[6] Gordon, C. McA.. Dehn surgery and satellite knots. Trans. Amer. Math. Soc. 275 (1983), 687–708.CrossRefGoogle Scholar
[7] Gordon, C. McA. and Litherland, R. A.. Incompressible planar surfaces in 3-manifolds. Topology Appl. 18 (1984), 121–144.CrossRefGoogle Scholar
[8] Moser, L.. Elementary surgery along a torus knot. Pacific J. Math. 38 (1971), 737–745.CrossRefGoogle Scholar
[9] Rolfsen, D.. Knots and Links. Mathematics Lecture Series 7 (Publish or Perish Inc., 1976).Google Scholar