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Non-Orientable Surfaces and Dehn Surgeries

Published online by Cambridge University Press:  20 November 2018

D. Matignon  and
N. Sayari
D. Matignon
Affiliation:
Université d’Aix-Marseille I, C.M.I. 39, rue Joliot Curie, F-13453 Marseille Cedex 13, France e-mail: matignon@cmi.univ-mrs.fr
N. Sayari
Affiliation:
Université de Moncton, Département de Mathematiques et de Statistique, Moncton, New Brunswick, E1A 3E9 e-mail: sayarin@umoncton.ca
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Abstract

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Let $K$ be a knot in ${{S}^{3}}$ . This paper is devoted to Dehn surgeries which create 3-manifolds containing a closed non-orientable surface $\hat{S}$ . We look at the slope $p/q$ of the surgery, the Euler characteristic $\mathcal{X}(\hat{S})$ of the surface and the intersection number $s$ between $\hat{S}$ and the core of the Dehn surgery. We prove that if $\mathcal{X}(\hat{S})\,\ge \,15\,-3q$, then $s\,=\,1$. Furthermore, if $s\,=\,1$ then $q\,\le \,4\,-\,3\,\mathcal{X}(\hat{S})$ or $K$ is cabled and $q\,\le \,8\,-5\mathcal{X}(\hat{S})$ . As consequence, if $K$ is hyperbolic and $\mathcal{X}(\hat{S})\,=\,-1$ , then $q\,\le \,7$.

Keywords

57M2557N1057M15Non-orientable surfaceDehn surgeryIntersection graphs
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 5 , 01 October 2004 , pp. 1022 - 1033
Copyright
Copyright © Canadian Mathematical Society 2004

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