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# Non-Orientable Surfaces and Dehn Surgeries

Published:2004-10-01
Printed: Oct 2004
• D. Matignon
• N. Sayari
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## Abstract

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 $\ch S$. We look at the slope ${p}/{q}$ of the surgery, the Euler characteristic $\chi(\ch S)$ of the surface and the intersection number $s$ between $\ch S$ and the core of the Dehn surgery. We prove that if $\chi(\hat S) \geq 15 - 3q$, then $s=1$. Furthermore, if $s=1$ then $q\leq 4-3\chi(\ch S)$ or $K$ is cabled and $q\leq 8-5\chi(\ch S)$. As consequence, if $K$ is hyperbolic and $\chi(\ch S)=-1$, then $q\leq 7$.
 Keywords: Non-orientable surface, Dehn surgery, Intersection graphs
 MSC Classifications: 57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57N10 - Topology of general $3$-manifolds [See also 57Mxx] 57M15 - Relations with graph theory [See also 05Cxx]