Non-hyperbolic Dehn surgeries on hyperbolic knots

双曲结的非双曲 Dehn 手术

• 批准号: 15540095
• 负责人: MOTEGI Kimihiko
• 金额: $ 1.28万
• 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: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540095/
• 关键词: hyperbolic knot; Dehn surgery; exceptional surgery; longitudinal surgery; fibered knot; surface bundle over the circle; Seifert fiber space; Nielsen-Thurston types; primitive; Seifert構成; 結ぴ目の対称性

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 counterexample 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 manifolds. 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.Dean introduced a primitive/Seifert-fibered construction, which is a natural modification of Berge's construction and provides infinite families of knots each of which admits Seifert fibered surgery. We determine non-hyperbolic, primitive/Seifert-fibered knots and show that for each such knots any integral, small Seifert surgery arises from a primitive/Seifert-fibered construction. We also study longitudinal exceptional surgeries on hyperbolic knots.Gabai found a hyperbolic, fibered knot in the 3-sphere on which a longitudinal surgery produces a toroidal manifold, and now it is known that there are infinitely many such hyperbolic, fibered knots. On the other hand, there have been no known examples of hyperbolic, fibered knots with longitudinal, Seifert fibered surgeries, and Teragaito asks if there are no such examples. We give an answer this question by constructing an infinite family of hyperbolic, fibered knots each of which admits a longitudinal, Seifert fibered surgery.

Thurston的双曲Dehn手术定理断言，如果3-球面中的纽结K是双曲的（即，K的补允许有限体积的完全双曲结构），则K上几乎所有的Dehn运算都产生双曲3-流形。然后，重要的是描述双曲结上的非双曲手术。众所周知，任何非双曲线手术都是缩小手术、环形手术、塞弗特纤维手术或产生几何化猜想的反例的手术。环面纽结上的塞弗特纤维化手术可以通过考虑外部的塞弗特纤维化如何在surgered流形上延伸来自然地解释。有没有任何自然的解释塞弗特纤维手术的双曲结？Berge给出了一个明确的建设，产生几个无限的家庭结每个承认一个透镜空间Dehn手术。迪恩介绍了一个原始/塞弗特纤维建设，这是一个自然的修改Berge的建设，并提供无限的家庭结每个承认塞弗特纤维手术。我们确定了非双曲线的、原始的/塞弗特纤维的结，并表明对于每个这样的结，任何完整的、小的塞弗特手术都来自原始的/塞弗特纤维的结构。我们还研究了双曲结的纵向例外手术。Gabai在3-球面上发现了一个双曲的纤维结，在这个球面上纵向手术产生了一个环面流形，现在我们知道有无限多个这样的双曲的纤维结。另一方面，还没有已知的双曲、纤维结与纵向、塞弗特纤维手术的例子，寺街人问是否没有这样的例子。我们给出了一个答案这个问题，通过构建一个无限家庭的双曲，纤维结，其中每个承认一个纵向，塞弗特纤维手术。

项目成果

Mohamed Ait-Nouh: "Obtaining graph knots by twisting unknots"Topology Appl.. (発表予定).

Mohamed Ait-Nouh：“通过扭转线结获得图结”拓扑应用程序..（待提交）。

Kimihiko Motegi: "An experimental study of Seifert fibered Dehn surgery via SnapPea"Interdisciplinary Information Sciences. 9. 95-125 (2003)

Kimihiko Motegi：“通过 SnapPea 对 Seifert 纤维 Dehn 手术进行的实验研究”跨学科信息科学。

Mohamed Ait-Nouh: "Twisted unknots"C.R.Acadi.Sci.Paris, Ser.I. 337. 321-326 (2003)

Mohamed Ait-Nouh：“扭曲的结”C.R.Acadi.Sci.Paris，Ser.I。

On primitive/Seifert-fibered constructions

关于原始/Seifert 纤维结构

• 期刊: Math.Proc.Camb.Phil.Soc. (印刷中)
• 作者: M.KADOWAKI;H.NAKAZAWA;K.WATANABE;Mituteru KADOWAKI;Mohamed Ait-Nouh;Kazuo MUKOYAMA;Mohamed Ait Nouh;Kimihiko Motegi;M.Aitnouh;Kazuhiro Ichihara;K.Miyazaki
• 通讯作者: K.Miyazaki

Braids and Nielsen-Thurston types of automorphisms of punctured surfaces

刺穿表面的辫子和 Nielsen-Thurston 类型的自同构

• 期刊: Tokyo Journal of Mathematics (to appear)
• 作者: Kobayashi;Teiichi;Minyou Katagiri;T.Shibata;Tsuyoshi Kobayashi;T.Kobayashi;Tsuyoshi Kobayashi;S.Matsumoto;T.Inaba;Kazuhiro Ichihara;Tsuyoshi Kobayashi;Kazuhiro Ichihara;Tsuyoshi Kobayashi;S.Matsumoto;Kazuhiro Ichihara;Takashi Inaba;T.Shibata;Tsuyoshi Kobayashi;T.Shibata;Kazuhiro Ichihara
• 通讯作者: Kazuhiro Ichihara