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Uniqueness of bridge surfaces for 2-bridge knots

Published online by Cambridge University Press:  01 May 2008

MARTIN SCHARLEMANN  and
MAGGY TOMOVA
MARTIN SCHARLEMANN*
Affiliation:
Mathematics Department, University of California Santa Barbara, Santa Barbara, CA 93117, U.S.A. email: [email protected]
MAGGY TOMOVA
Affiliation:
Mathematics Department, Rice University, 6100 S. Main St., Houston TX 77005-1892, U.S.A. email: [email protected]
*
†Partially supported by a National Science Foundation grant.
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Abstract

Any 2-bridge knot in S3 has a bridge sphere from which any other bridge surface can be obtained by stabilization, meridional stabilization, perturbation and proper isotopy.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 3 , May 2008 , pp. 639 - 650
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[BoO]Bonahon, F. and Otal, J. P.. Scindements de Heegaard des espaces lenticulaires. Ann. Sci. École. Norm. Sup. 16 (1983), 451–466.CrossRefGoogle Scholar
[BZ]Boileau, M. and Zieschang, H.. Heegaard genus of closed orientable Seifert 3-manifolds. Invent. Math. 76 (1984), 455–468.CrossRefGoogle Scholar
[CG]Casson, A. and Gordon, C. McA.. Reducing Heegaard splittings. Topology Appl. 27 (1987), 275–283.CrossRefGoogle Scholar
[GST]Goda, H., Scharlemann, M. and Thompson, A.. Levelling an unknotting tunnel. Geom. Topol. 4 (2000), 243–275.CrossRefGoogle Scholar
[HT]Hatcher, A. and Thurston, W.. Incompressible surfaces in 2-bridge knot complements. Invent. Math. 79 (1985), 225–246.CrossRefGoogle Scholar
[HS2]Hayashi, C. and Shimokawa, K.. Heegaard splittings of trivial arcs in compression bodies. J. Knot Theory Ramifications 10 (2001), 71–87.CrossRefGoogle Scholar
[Ko1]Kobayashi, T.. Classification of unknotting tunnels for two bridge knots. In Proceedings of the Kirbyfest Geometry and Topology Monographs 2 (1999), 259–290.CrossRefGoogle Scholar
[Ko2]Kobayashi, T.. Heegaard splittings of exteriors of two bridge knots. Geom. Topol. 5 (2001), 609–650.CrossRefGoogle Scholar
[MS]Moriah, Y. and Schultens, J.. Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal. Topology 37 (1998), 1089–1112.CrossRefGoogle Scholar
[OS]Ozsv, P. and Szab, Z.. An introduction to Heegaard Floer homology. Floer homology, gauge theory, and low-dimensional topology. Clay Math. Proc. 5 (2006), 3–27.Google Scholar
[Ot1]Otal, J.-P.. Presentations en ponts du nœud trivial. C. R. Acad. Sci. Paris Sr. I Math. 294 (1982), 553–556.Google Scholar
[Ot2]Otal, J.-P.. Presentations en ponts des nœud rationnels. Low-dimensional topology London. Math. Soc. Lecture Note Ser. 95 (1985), 143–160.Google Scholar
[RS1]Rubinstein, H. and Scharlemann, M.. Comparing Heegaard splittings of non-Haken 3-manifolds. Topology 35 (1997), 1005–1026.CrossRefGoogle Scholar
[RS2]Rubinstein, H. and Scharlemann, M.. Comparing Heegaard splittings of non-Haken 3-manifolds – the bounded case. Trans. Amer. Math. Soc. 350 (1998), 689–715.CrossRefGoogle Scholar
[STo]Scharlemann, M. and Tomova, M.. Conway products and links with multiple bridge surfaces. ArXiv preprint math.GT/0608435.Google Scholar
[To]Tomova, M.. Thin position for knots in a 3-manifold. ArXiv preprint math.GT/0609674.Google Scholar
[Wa]Waldhausen, F.. Heegaard-Zerlegungen der 3-Sphäre. Topology 7 (1968), 195–203.CrossRefGoogle Scholar
[Wi]Witten, E.. Topological quantum field theory. Comm. Math. Phys. 117 (1988), 353–386.CrossRefGoogle Scholar