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Closed incompressible surfaces in 2-generator hyperbolic 3-manifolds with a single cusp

Published online by Cambridge University Press:  20 January 2009

D. D. Long  and
A. W. Reid
D. D. Long
Affiliation:
Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
A. W. Reid
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA
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Abstract

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A knot K is said to have tunnel number 1 if there is an embedded arc A in S3, with endpoints on K, whose interior is disjoint from K and such that the complement of a regular neighbourhood of K ∪ A is a genus 2 handlebody. In particular the fundamental group of the complement of a tunnel number one knot is 2-generator. There has been some interest in the question as to whether there exists a hyperbolic tunnel number one knot whose complement contains a closed essential surface. The aim of this paper is to prove the existence of infinitely many 2-generator hyperbolic 3-manifolds with a single cusp which contain a closed essential surface. One such example is a knot complement in RP3. The methods used are of interest as they include the possibility that one of our examples is a knot complement in S3.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 3 , October 1993 , pp. 501 - 513
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

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