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Uniqueness of higher genus bridge surfaces for torus knots

Published online by Cambridge University Press:  11 May 2015

ALEXANDER ZUPAN
ALEXANDER ZUPAN*
Affiliation:
Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX 78712, U.S.A. e-mail: [email protected]
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Abstract

We show that a torus knot which is not 2-bridge has a unique irreducible bridge splitting of positive genus.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 1 , July 2015 , pp. 79 - 88
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[1] Bachman, D. and Schleimer, S. Distance and bridge position. Pacific J. Math. 219 (2005), 221–235.Google Scholar
[2] Baker, K., Gordon, C. McA. and Luecke, J. Bridge number, Heegaard genus and non-integral Dehn surgery. Preprint, arXiv:1202.0263.Google Scholar
[3] Blair, R., Campisi, M., Johnson, J., Taylor, S. and Tomova, M. Exceptional and cosmetic surgeries on knots. Preprint, arXiv:1209.0197.Google Scholar
[4] Boileau, M., Rost, M. and Zieschang, H. On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces. Math. Ann. 279 (1988), 553–581.Google Scholar
[5] Hayashi, C. and Shimokawa, K. Heegaard splittings of trivial arcs in compression bodies. J. Knot Theory Ramifications 10 (2001), 71–87.Google Scholar
[6] Jang, Y. Three-bridge links with infinitely many three-bridge spheres. Topology Appl. 157 (2010), 165–172.Google Scholar
[7] Moriah, Y. Heegaard splittings of Seifert fibered spaces. Invent. Math. 91 (1988), 465–481.Google Scholar
[8] Moriah, Y. and Schultens, J. Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal. Topology 37 (1998), 1089–1112.Google Scholar
[9] Morimoto, K. On minimum genus Heegaard splittings of some orientable closed 3-manifolds. Tokyo J. Math. 12 (1989), 321–355.Google Scholar
[10] Otal, J.-P. Presentations en ponts des noeuds trivial. C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 553–556.Google Scholar
[11] Ozawa, M. Nonminimal bridge positions of torus knots are stabilized. Math. Proc. Camb. Phil. Soc. 151 (2011), 307–317.Google Scholar
[12] Ozawa, M. and Takao, K. A locally minimal, but not globally minimal bridge position of a knot. Math. Proc. Camb. Phil. Soc. 155 (2013), 181–190.Google Scholar
[13] Scharlemann, M. and Tomova, M. Uniqueness of bridge surfaces for 2-bridge knots. Math. Proc. Camb. Phil. Soc. 144 (2008), 639–650.Google Scholar
[14] Schubert, H. Über eine numerische Knoteninvariante. Math. Z. 61 (1954), 245–288.Google Scholar
[15] Schultens, J. Bridge numbers of torus knots. Math. Proc. Camb. Phil. Soc. 143 (2007), 621–625.Google Scholar
[16] Tomova, M. Thin position for knots in a 3-manifold. J. London Math. Soc. (2) 80 (2009), 85–98.Google Scholar
[17] Zupan, A. Properties of knots preserved by cabling. Comm. Anal. Geom. 19 (2011), 541–562.Google Scholar
[18] Zupan, A. Bridge and pants complexities of knots. J. London Math. Soc. (2) 87 (2013), 43–68.Google Scholar
[19] Zupan, A. Bridge spectra of iterated torus knots. Comm. Anal. Geom. 22 (2014), 931–963.Google Scholar