arXiv 1109.51512011-09math.GT

含有本质一次穿孔环面的纽结流形的Dehn填充

Dehn fillings of knot manifolds containing essential once-punctured tori

Steven Boyer, Cameron McA. Gordon, Xingru Zhang

备注：: 55 pages, 41 figures. Minor changes incorporating the referee's comments, to appear in Transactions of AMS

摘要：

本文研究了双曲纽结流形上的例外Dehn填充，该流形包含一个本质的一次穿孔环面。令$M$是这样一个纽结流形，并令$β$是这样一个基本的一次穿孔环面的边界斜率。证明了若Dehn用斜率$α$填充$M$产生一个Seifert流形，则$Δ(α,β)\leq 5$ .此外，当$\D(α,β)\geq 4$时，我们对三元组$(M; α,β)$进行分类。更准确地说，当$\D(α,β)=5$，则$M$是（唯一的）流形$Wh(-3/2)$，通过Dehn用斜率-3/2填充Whitehead链路外部的一个边界分量而获得，并且$(α, β)$是斜率对$(-5, 0)$。此外，$\D(α,β)=4$当且仅当$(M; α,β)$是三元组$\displaystyle (Wh(\frac{-2n\pm1}{n}); -4, 0)$，对于某个整数$n$和$|n|>1$。结合这与已知的结果，我们分类所有双曲结流形$M$和对斜坡$(β, γ)$ $\partial M$，其中$β$是边界斜坡的一个重要的一次穿刺环面$M$和$γ$是一个例外的填充斜坡的距离4或更多的$β$。在$S^3$中给出了双曲亏格单结外点特殊情况下的精化结果。

英文摘要：

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 $β$ be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling $M$ with slope $α$ produces a Seifert fibred manifold, then $Δ(α,β)\leq 5$. Furthermore we classify the triples $(M; α,β)$ when $\D(α,β)\geq 4$. More precisely, when $\D(α,β)=5$, 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 $(α, β)$ is the pair of slopes $(-5, 0)$. Further, $\D(α,β)=4$ if and only if $(M; α,β)$ 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 $(β, γ)$ on $\partial M$ where $β$ is the boundary slope of an essential once-punctured torus in $M$ and $γ$ is an exceptional filling slope of distance 4 or more from $β$. Refined results in the special case of hyperbolic genus one knot exteriors in $S^3$ are also given.

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