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Hyperbolic knot complements without closed embedded totally geodesic surfaces

Part of: Low-dimensional topology

Published online by Cambridge University Press:  09 April 2009

Kazuhiro Ichihara  and
Makoto Ozawa
Kazuhiro Ichihara
Affiliation:
Department of Mathematics Tokyo Institute of TechnologyOokayama 2-12-1, Meguroku Tokyo 152-8551Japan e-mail: ichihara@math.titech.ac.jp
Makoto Ozawa
Affiliation:
Department of Mathematics School of Education Waseda UniversityNishiwaseda 1-6-1, Shinjuku-ku Tokyo 169-8050Japan e-mail: ozawa@mn. waseda.ac.jp
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Abstract

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It is conjectured that a hyperbolic knot complement does not contain a closed embedded totally geodesic surface. In this paper, we show that there are no such surfaces in the complements of hyperbolic 3-bridge knots and double torus knots. Some topological criteria for a closed essential surface failing to be totally geodesic are given. Roughly speaking, sufficiently ‘complicated’ surfaces cannot be totally geodesic.

Keywords

accidental peripheraldouble torustotally geodesic surface

MSC classification

Secondary: 57M25: Knots and links in $S^3$ 57M50: Geometric structures on low-dimensional manifolds
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 3 , June 2000 , pp. 379 - 386
Copyright
Copyright © Australian Mathematical Society 2000

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