Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-29T19:11:54.875Z Has data issue: false hasContentIssue false

Uniqueness of higher genus bridge surfaces for torus knots

Published online by Cambridge University Press:  11 May 2015

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

Abstract

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

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bachman, D. and Schleimer, S. Distance and bridge position. Pacific J. Math. 219 (2005), 221235.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), 553581.Google Scholar
[5] Hayashi, C. and Shimokawa, K. Heegaard splittings of trivial arcs in compression bodies. J. Knot Theory Ramifications 10 (2001), 7187.Google Scholar
[6] Jang, Y. Three-bridge links with infinitely many three-bridge spheres. Topology Appl. 157 (2010), 165172.Google Scholar
[7] Moriah, Y. Heegaard splittings of Seifert fibered spaces. Invent. Math. 91 (1988), 465481.Google Scholar
[8] Moriah, Y. and Schultens, J. Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal. Topology 37 (1998), 10891112.Google Scholar
[9] Morimoto, K. On minimum genus Heegaard splittings of some orientable closed 3-manifolds. Tokyo J. Math. 12 (1989), 321355.Google Scholar
[10] Otal, J.-P. Presentations en ponts des noeuds trivial. C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 553556.Google Scholar
[11] Ozawa, M. Nonminimal bridge positions of torus knots are stabilized. Math. Proc. Camb. Phil. Soc. 151 (2011), 307317.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), 181190.Google Scholar
[13] Scharlemann, M. and Tomova, M. Uniqueness of bridge surfaces for 2-bridge knots. Math. Proc. Camb. Phil. Soc. 144 (2008), 639650.Google Scholar
[14] Schubert, H. Über eine numerische Knoteninvariante. Math. Z. 61 (1954), 245288.Google Scholar
[15] Schultens, J. Bridge numbers of torus knots. Math. Proc. Camb. Phil. Soc. 143 (2007), 621625.Google Scholar
[16] Tomova, M. Thin position for knots in a 3-manifold. J. London Math. Soc. (2) 80 (2009), 8598.Google Scholar
[17] Zupan, A. Properties of knots preserved by cabling. Comm. Anal. Geom. 19 (2011), 541562.Google Scholar
[18] Zupan, A. Bridge and pants complexities of knots. J. London Math. Soc. (2) 87 (2013), 4368.Google Scholar
[19] Zupan, A. Bridge spectra of iterated torus knots. Comm. Anal. Geom. 22 (2014), 931963.Google Scholar