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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-球面上的一个纽结K是双曲的(即，K的补集允许有一个有限体积的完全双曲结构)，则K上的所有Dehn手术都产生双曲3-流形。然后，描述双曲纽结上的非双曲手术是很重要的。众所周知，任何非双曲手术都是一种约简手术、环状手术、Seifert纤维手术或产生反例几何猜想的手术。在本研究中，我们主要研究Seifert纤维手术。Seifert纤维环结手术可以通过考虑Seifert外部纤维如何延伸到手术歧管上来自然地解释。Seifert光纤手术在双曲线结上有什么自然的解释吗？Berge给出了一个显式构造，它产生了几个无穷族纽结，每个纽结允许一个透镜空间Dehn手术。据推测，任何晶状体间隙手术都可以是…更多的解释是由Berge的原语/原语结构解释的。Dean完成了原始/原始结构的自然推广，这解释了许多双曲线结上的Seifert纤维外科手术。1996年，戈登推测，任何塞弗特纤维手术都可以用迪恩的原始/塞弗特纤维结构来解释。最近，尤达夫-穆尼奥斯已经证明，所有已知的由孟德西诺斯技巧构建的塞弗特纤维手术的例子都可以用原始/塞弗特纤维结构来解释。在这项研究中，作为Thomas Mattman和Katura Miyazaki的共同工作，我们构造了两个无穷族纽结，每个纽结都接受Seifert纤维手术，这些手术没有一个来自Dean的S原始/Seifert纤维结构。这推翻了一个猜想，即所有的Seifert纤维外科手术都来自Dean‘S原始/Seifert纤维结构。(-3，3，5)-椒盐卷曲纽结属于两个无穷族。最近我们感兴趣的问题是：如果K有Seifert纤维外科手术，那么K是否嵌入到3-球面的亏格2 Heegaard曲面中？较少