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Non-hyperbolic Dehn surgeries on hyperbolic knots

双曲结的非双曲 Dehn 手术

基本信息

批准号：
13640089
负责人：
MOTEGI Kimihiko
金额：
$ 1.15万
依托单位：
Nihon university
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640089/
关键词：
Dehn surgery hyperbolic knot Seifert fiber space primitive Seifert construction symmetry of knots tunnel number primitive toroidal manifold seifert-fibered construction

项目摘要

Thurston's hyperbolic Dehn surgery theorem asserts that if a knot K in the 3-sphere is hyperbolic (i.e., the complement of K admits a complete hyperbolic structure of finite volume), then all but finitely many Dehn surgeries on K yield hyperbolic 3-manifolds. Then it is important to describe non-hyperbolic surgeries on hyperbolic knots. It is known that any non-hyperbolic surgery is a reducing surgery, a toroidal surgery, a Seifert fibered surgery, or a surgery producing a counter example to Geometrization conjecture.In this research we focus on Seifert fibered surgeries. Seifert fibered surgeries on torus knots can be naturally explained by considering how Seifert fibrations of the exterior extends over the surgered manifold. Are there any natural explanation for Seifert fibered surgeries on hyperbolic knots? Berge gave an explicit construction which yields several infinite families of knots each admitting a lens space Dehn surgery. It is conjectured that any lens space surgery can be … More explained by Berge's primitive/primitive construction. The natural generalization of primitive /primitive construction was done by Dean, which explains many Seifert fibered surgeries on hyperbolic knots. In 1996, Gordon conjectured that any Seifert fibered surgery can be explained by Dean's primitive/Seifert-fibered construction. Recently Eudave-Munoz has shown that all known examples of Seifert fibered surgeries constructed by the Montesinos trick can be explained by primitive/Seifert-fibered construction. In this research, as a joint work with Thomas Mattman and Katura Miyazaki, we construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean' s primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert fibered surgeries arise from Dean' s primitive/Seifert-fibered construction. The (-3, 3, 5)-pretzel knot belongs to both of the infinite families. Very recently we are interested in the questions : If K has a Seifert fibered surgery, then is K embedded in a genus 2 Heegaard surface of the 3-sphere? Less

瑟斯顿的双曲 Dehn 手术定理断言，如果 3-球体中的结 K 是双曲的（即 K 的补集承认有限体积的完全双曲结构），那么除了有限多个 Dehn 手术之外，所有 K 上的手术都会产生双曲 3-流形。那么描述双曲线结上的非双曲线手术就很重要。众所周知，任何非双曲线手术都是缩小手术、环形手术、Seifert纤维手术或产生几何化猜想反例的手术。在本研究中，我们重点关注Seifert纤维手术。环面结上的 Seifert 纤维手术可以通过考虑外部 Seifert 纤维如何延伸到手术歧管上来自然地解释。对于双曲线结上的 Seifert 纤维手术有任何自然的解释吗？ Berge 给出了一个明确的构造，它产生了几个无限的结族，每个结都接受透镜空间 Dehn 手术。据推测，任何晶状体空间手术都可以用伯格的原始/原始构造来解释。原始/原始构造的自然概括是由 Dean 完成的，这解释了许多关于双曲结的 Seifert 纤维手术。 1996 年，戈登推测任何 Seifert 纤维手术都可以用 Dean 的原始/Seifert 纤维结构来解释。最近，Eudave-Munoz 表明，所有已知的由 Montesinos 技巧构建的 Seifert 纤维手术的例子都可以用原始/Seifert 纤维结构来解释。在这项研究中，作为与 Thomas Mattman 和 Katura Miyazaki 的合作，我们构建了两个无限的结族，每个结都允许进行 Seifert 纤维手术，而这些手术中没有一个来自 Dean 的原始/Seifert 纤维结构。这反驳了所有 Seifert 纤维手术都源自 Dean 的原始/Seifert 纤维结构的猜想。 (-3, 3, 5)-椒盐卷饼结属于这两个无限家族。最近我们对以下问题感兴趣：如果 K 进行了 Seifert 纤维手术，那么 K 是否嵌入 3 球体的 2 Heegaard 表面？较少的