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Dehn surgery and essential annuli*

Published online by Cambridge University Press:  24 October 2008

Chuichiro Hayashi
Chuichiro Hayashi
Affiliation:
Department of Science, Gakushuin University, Mejiro, Tokyo 171, Japan
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Extract

In this paper we consider Dehn surgery and essential annuli whose two boundary components are in distinct components of the boundary of a 3-manifold.

Let Nl be an orientable 3-manifold with boundary, Kl a knot in Nl, and N2 the 3-manifold obtained by performing γ-Dehn surgery Kl. In detail, let Vl be a regular neighbourhood Kl, X = Nl − int Vl the exterior of Kl, T the toral component ∂Vl of ∂X, and γ a slope on T. Then we obtain the 3-manifold N2 by attaching a solid torus V2 to X so that γ bounds a disc in V2. Let K2 be the core of V2. Let π be the slope of a meridian loop of Kl, and Δ the distance between the slopes π and γ, i.e. the minimal number of intersection points of the two slopes on T. Suppose for i = 1 and 2 that Ni contains a proper annulus Ai such that the two components of ∂Ai are essential loops on distinct incompressible components of ∂Ni. Then note that Ai is essential, i.e. incompressible and ∂-incompressible in Ni.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 1 , July 1996 , pp. 127 - 146
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

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