论文标题
$ s^3 $与适当的功率曲线的隧道单个结外部
Tunnel-number-one knot exteriors in $S^3$ disjoint from proper power curves
论文作者
论文摘要
正如双曲原始/塞弗特结的分类项目之一的背景论文之一，其完整列表在[BK20]中给出，本文分类了所有可能的R-R图，这是两个不连接的简单封闭曲线$ r $和$β$的所有可能的R-r图，并在$β$ upe $β$等等的边界上均与$β$ upe curve and up curve h h h he嵌入$ s^3 $，因此$ h [r] $是隧道单个结的外部。 As a consequence, if $R$ is a nonseparating simple closed curve on the boundary of a genus two handlebody such that $H[R]$ embeds in $S^3$, then there exists a proper power curve disjoint from $R$ if and only if $H[R]$ is the exterior of the unknot, a torus knot, or a tunnel-number-one cable of a torus knot. 本文的结果将主要用于证明p/sf结的双曲性，并在$ s^3 $中曾经是p/sf结的p/sf节分类，这是[bk20]中的p/sf节的类型之一。与这些结果一起，本文由三个部分组成：三个图表：Heegaard图，R-R图和混合图，“剔除引理”，以及将波浪定位到R-R图中，也将用于R-R-R图，也将用于分类的超纤维主/Seifert/Seifert/Seifert Knots in $ s $ s $ s $ s^3 $^3 $^3 $。
As one of the background papers of the classification project of hyperbolic primitive/Seifert knots in $S^3$ whose complete list is given in [BK20], this paper classifies all possible R-R diagrams of two disjoint simple closed curves $R$ and $β$ lying in the boundary of a genus two handlebody $H$ up to equivalence such that $β$ is a proper power curve and a 2-handle addition $H[R]$ along $R$ embeds in $S^3$ so that $H[R]$ is the exterior of a tunnel-number-one knot. As a consequence, if $R$ is a nonseparating simple closed curve on the boundary of a genus two handlebody such that $H[R]$ embeds in $S^3$, then there exists a proper power curve disjoint from $R$ if and only if $H[R]$ is the exterior of the unknot, a torus knot, or a tunnel-number-one cable of a torus knot. The results of this paper will be mainly used in proving the hyperbolicity of P/SF knots and in classifying P/SF knots in once-punctured tori in $S^3$, which is one of the types of P/SF knots in [BK20]. Together with these results, the preliminary of this paper which consists of three parts: the three diagrams which are Heegaard diagrams, R-R diagrams, and hybrid diagrams, `the Culling Lemma', and locating waves into an R-R diagrams, will also be used in the classification of hyperbolic primitive/Seifert knots in $S^3$.
