Non-Orientable Surfaces and Dehn Surgeries

Matignon, D. ; Sayari, N.

Canadian journal of mathematics, Tome 56 (2004) no. 5, pp. 1022-1033

Résumé

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 $q\,\le \,7$ .

Détail

DOI : 10.4153/CJM-2004-046-9

Mots-clés : 57M25, 57N10, 57M15, Non-orientable surface, Dehn surgery, Intersection graphs

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