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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

References

[1]Adams, C., ‘Toroidally alternating knots and links’, Topology 33 (1994), 353–369.CrossRefGoogle Scholar
[2]Adams, C., Brock, J., Bugbee, J., Comar, T., Faigin, K., Joseph, A. and Pesikoff, D., ‘Almost alternating links’, Topology Appl. 46 (1992), 151–165.CrossRefGoogle Scholar
[3]Adams, C. and Reid, A., ‘Quasi-Fuchsian surfaces in hyperbolic knot complements’, J. Austral. Math. Soc. (Series A) 55 (1993), 116–131.CrossRefGoogle Scholar
[4]Brittenham, M., ‘Bounding canonical genus bounds volume’, preprint available on http://xxx.lanl.gov/abs/math.GT/9809142.Google Scholar
[5]Kirby, R., ‘Problems in low-dimensional topology’, in: Geometric topology, Part 2 (ed. Kazez, W. H.), Studies in Adv. Math. (Amer. Math. Soc., Inters. Press, 1997).Google Scholar
[6]Lozano, M. T. and Przytycki, J. H., ‘Incompressible surfaces in the exterior of a closed 3-braid l, surfaces with horizontal boundary components’, Math. Proc. Cambridge Philos. Soc. 98 (1985), 275–299.CrossRefGoogle Scholar
[7]Menasco, W., ‘Closed incompressible surfaces in alternating knot and link complements’, Topology 23 (1984), 37–44.CrossRefGoogle Scholar
[8]Miyamoto, Y., ‘Volumes of hyperbolic manifolds with geodesic boundary’, Topology 33 (1994), 613–629.CrossRefGoogle Scholar
[9]Menasco, W. and Reid, A., ‘Totally geodesic surfaces in hyperbolic link complements’, in: Topology'90 (eds. Apanasov, B., Neumann, W., Reid, A. and Siebenmann, L.) (de Gruyter, Amsterdam, 1992).Google Scholar
[10]Oertel, U., ‘Closed incompressible surfaces in complements of star links’, Pacific J. Math. 111 (1984), 209–230.CrossRefGoogle Scholar
[11]Ozawa, M., ‘Synchronism of an incompressible non-free Seifert surface for a knot and an algebraically split closed surface in the knot complement’, to appear in Proc. Amer. Math. Soc..Google Scholar