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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

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