论文标题
双曲线隧道单数结，带有塞弗特纤维的dehn手术
Hyperbolic tunnel-number-one knots with Seifert-fibered Dehn surgeries
论文作者
论文摘要
Suppose $α$ and $R$ are disjoint simple closed curves in the boundary of a genus two handlebody $H$ such that $H[R]$ embeds in $S^3$ as the exterior of a hyperbolic knot $k$(thus, $k$ is a tunnel-number-one knot), and $α$ is Seifert in $H$(i.e., a 2-handle addition $H[α]$ is a Seifert纤维空间），而不是$ h [r] $的子午线。然后，对于由$α$代表的斜率$γ$的$ k $，$γ$ -Dehn手术$ K（γ）$是塞弗特纤维的空间。 Seifert纤维纤维的Dehn手术的这种结构概括了由结的启动/塞菲尔特位置引起的Seifert纤维纤维手术，该手术是在[D03]中引入的。 在本文中，我们表明存在$ \ partial h $中的子午曲线$ m $ $ k $（或$ h [r] $），因此$α$与$ m $相交的$ m $恰好在一个点上横断。因此，这种结构是由塞弗特（Seifert）纤维的Dehn手术$ K（γ）$构建，可以是由$ k $的启动/塞法特（Seifert）位置产生的，其表面坡度为$γ$。该结果部分支持了两个猜想：（1）$ s^3 $中双曲线结上的任何塞菲尔特纤维纤维手术都是不可或缺的，并且（2）（2）在双曲线隧道隧道单个结上进行的任何塞菲尔特纤维纤维手术。
Suppose $α$ and $R$ are disjoint simple closed curves in the boundary of a genus two handlebody $H$ such that $H[R]$ embeds in $S^3$ as the exterior of a hyperbolic knot $k$(thus, $k$ is a tunnel-number-one knot), and $α$ is Seifert in $H$(i.e., a 2-handle addition $H[α]$ is a Seifert-fibered space) and not the meridian of $H[R]$. Then for a slope $γ$ of $k$ represented by $α$, $γ$-Dehn surgery $k(γ)$ is a Seifert-fibered space. Such a construction of Seifert-fibered Dehn surgeries generalizes that of Seifert-fibered Dehn surgeries arising from primtive/Seifert positions of a knot, which was introduced in [D03]. In this paper, we show that there exists a meridional curve $M$ of $k$ (or $H[R]$) in $\partial H$ such that $α$ intersects $M$ transversely in exactly one point. It follows that such a construction of a Seifert-fibered Dehn surgery $k(γ)$ can arise from a primtive/Seifert position of $k$ with $γ$ its surface-slope. This result supports partially the two conjectures: (1) any Seifert-fibered surgery on a hyperbolic knot in $S^3$ is integral, and (2) any Seifert-fibered surgery on a hyperbolic tunnel-number-one knot arises from a primitive/Seifert position whose surface slope corresponds to the surgery slope.
