Mathematics > Geometric Topology
[Submitted on 23 Sep 2011 (v1), last revised 23 Mar 2012 (this version, v3)]
Title:Dehn fillings of knot manifolds containing essential once-punctured tori
View PDFAbstract:In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let $M$ be such a knot manifold and let $\beta$ be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling $M$ with slope $\alpha$ produces a Seifert fibred manifold, then $\Delta(\alpha,\beta)\leq 5$. Furthermore we classify the triples $(M; \alpha,\beta)$ when $\D(\alpha,\beta)\geq 4$. More precisely, when $\D(\alpha,\beta)=5$, then $M$ is the (unique) manifold $Wh(-3/2)$ obtained by Dehn filling one boundary component of the Whitehead link exterior with slope -3/2, and $(\alpha, \beta)$ is the pair of slopes $(-5, 0)$. Further, $\D(\alpha,\beta)=4$ if and only if $(M; \alpha,\beta)$ is the triple $\displaystyle (Wh(\frac{-2n\pm1}{n}); -4, 0)$ for some integer $n$ with $|n|>1$. Combining this with known results, we classify all hyperbolic knot manifolds $M$ and pairs of slopes $(\beta, \gamma)$ on $\partial M$ where $\beta$ is the boundary slope of an essential once-punctured torus in $M$ and $\gamma$ is an exceptional filling slope of distance 4 or more from $\beta$. Refined results in the special case of hyperbolic genus one knot exteriors in $S^3$ are also given.
Submission history
From: Xingru Zhang [view email][v1] Fri, 23 Sep 2011 18:26:16 UTC (1,159 KB)
[v2] Mon, 24 Oct 2011 17:02:57 UTC (1,321 KB)
[v3] Fri, 23 Mar 2012 21:24:47 UTC (1,321 KB)
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