Geometric structures of 3-manifolds and various related structures

三流形的几何结构及各种相关结构

• 批准号: 17540077
• 负责人: KOBAYASHI Tsuyoshi
• 金额: $ 1.41万
• 依托单位: Nara Women's University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540077/
• 关键词: knot; tunnel number; Seifert surface; automatic group; growth function; Heegaard splitting; 結び目、絡み目; 橋指数; Heegaard分解; Seifert曲面; plumbing

In this research project, we obtained the following results.1. We defined a numerical invariant, called growth rate of tunnel numbers, of knots in 3-manifolds. For m-small knots, we obtained the following.Suppose K is a m-small knot in. a 3-manifold M. Let g = g(X)-g(M), and b_i (i =1,..., g) be the bridge index of K with respect to genus g(X) - i Heegaard surface of M. Then the growth rate of K is given by min_i=_<1,..., n>{1-i/(b_i)}.2. Heegaard splittings of exteriors of knots.・ Let K_1, K_2 be knots in 3-manifolds, and T_1,T_2 tunnel systems of K_1, K_2 respectively. We gave a necessary and sufficient condition for the tunnel system t_1 ∪ T_2 of K_1#K_2 giving a stabilized Heegaard splitting.・ For each natural number n, there exists a knot K such that the equality g(nK) = gt(K) holds, where nK denotes the connected sum of n copies of K. This implies the existence of counterexample to Morimoto's Conjecture concerning super additive phenomina of tunnel number of knots.3. We showed that for any link L in the 3-sphere, there is a Seifert surface S for L such that S is obtained from a disk by successively plumbing flat annuli, where all of the attaching regions are contained in the disk.4. We made research on Gersten's Problem : each automatic group is either (1) a finite group, (2) contains a free abelian group of rank 2. or (3) a word hyperbolic group.We showed that for the n-starred automatic groups this assertion holds.5. Growth function of 2-bridge link groupsWe made computar experiments on the growth functions of 2-bridge link groups, and posed conjectures on the structure of the growth functions.

在本研究项目中，我们取得了以下成果。我们定义了三维流形中纽结的一个数值不变量，称为隧道数增长率。对于m-小结，我们得到了如下结果：设K是m-小结。设g=g(X)-g(M)，且bi(i=1，…，g)是K关于亏格g(X)-i Heegaard曲面的桥指数，则K的增长率由min_i=1，…，n&gt；{1-i/(Bi)}给出。结的外部的Heegaard分裂。设K_1，K_2是3-流形上的纽结，T_1，T_2分别是K_1，K_2的隧道系统。给出了K_1#K_2的隧道系统t_1-∪-T_2具有稳定Heegaard分裂的充要条件。对于每个自然数n，存在一个纽结K，使得等式g(NK)=gT(K)成立，其中NK表示K的n个副本的连通和。这意味着存在关于隧道节数的超加性现象的Morimoto猜想的反例。证明了对于3-球面上的任一环L，都有一个对应于L的塞弗特曲面S，使得S是通过连续地对平面环进行管道处理而从圆盘上获得的，其中所有的附着区都包含在圆盘中。我们研究了Gersten问题：每个自动机群是(1)一个有限群，(2)包含一个2阶的自由阿贝尔群，或(3)一个字双曲群。我们证明了对于n-星自动机群，这一断言成立。对两桥连接群的增长函数进行了计算实验，并对增长函数的结构提出了猜想。

项目成果

Essential laminations and branched surfaces in the exteriors of links

链节外部的基本叠片和分支表面

• 发表时间: 2005
• 期刊: Japan.J.Math 31
• 作者: M.Brittenham;C.Hayashi;M.Hirasawa;T.Kobayashi;K.Shimokawa
• 通讯作者: K.Shimokawa

On the growth rate of tunnel number of knots

论隧道节数增长率

• 期刊: Journal fur die Reine and Ang. Math. (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
• 通讯作者: Tsuyoshi Kobayashi

A search method for a thin position of a link

一种链接细位置的搜索方法

• 发表时间: 2005
• 期刊: Algebr. Geom. Topol. 5
• 作者: T.Kobayashi;Y.Riek;Yasushi Yamashita;Yamashita Yasushi;M.Brittenham;Tsuyoshi Kobayashi
• 通讯作者: Tsuyoshi Kobayashi

Drawing Bers embeddings of the Teichmu}ller space of once-punctured tori

绘制一次刺穿环面 Teichmuller 空间的 Bers 嵌入

• 期刊: Experimental Mathematics (to appear)
• 作者: Y.Komori;T.Sugawa;M.Wada;Y.Yamashita
• 通讯作者: Y.Yamashita

Heegaard genus of the connected sum of m-small knots

• DOI: 10.4310/cag.2006.v14.n5.a8
• 发表时间: 2005-03
• 期刊: arXiv: Geometric Topology
• 作者: Tsuyoshi Kobayashi;Y. Rieck