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的双曲线Dehn手术定理断言，如果三个球中的结K为双曲线（即，K的补体都可以接受有限体积的完整双曲线结构），则几乎有限的许多Dehn手术在K产量k屈服的三个手术中。然后，重要的是要在双曲结中描述非纤维化手术。众所周知，任何非纤维化手术都是减少手术，环形手术，塞菲特纤维手术或产生反示例构想的手术。通过考虑如何在手术歧管上延伸的Seifert纤维，可以自然解释圆环结上的Seifert纤维手术。双曲线结上塞弗特纤维手术是否有自然的解释？伯格（Berge）提供了明确的结构，该结构产生了几个无限的结系列，每个结构都承认了镜头空间Dehn手术。Dean引入了原始/塞菲尔特纤维的结构，这是对Berge结构的自然修改，并提供了无限的结族，每个结构都接受了Seifert纤维纤维手术。我们确定非纤维，原始/塞菲尔特纤维结的结，并表明，对于每个这样的结，任何积分，小的塞菲尔特手术都是由原始/塞菲尔特纤维的结构产生的。我们还研究了双曲线结的纵向特殊手术。Gabai在三个球体中发现了双曲线，纤维纤维结，纵向手术会产生旋风歧管，现在已经知道许多这样的双曲线纤维纤维结。另一方面，没有纵向，塞弗特纤维手术的双曲线，纤维结的示例，而Teragaito询问是否没有这样的例子。我们通过构建一个无限的双曲线，纤维结的家族来给出一个答案，每个纤维纤维结都接受了纵向，塞弗特纤维纤维手术。

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。

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

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