Geometric structures of 3-manifolds and various related structures

三流形的几何结构及各种相关结构

基本信息

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

项目摘要

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.

在这个研究项目中，我们进行了形式的不变性，称为M-Smart结的隧道数量折叠的增长率。（x）-g（m）和b_i（i = 1，...，g）是k的桥索引，相对于M g -i属。 {...，n> {1-i/（b_i）}。 ∪k_1＃k_2的T_2给出了稳定的Heegaard拆分，存在一个结，平等G（nk）= gt（k）持有，nk表示K关于打结的隧道数量的超级添加。3我们显示的任何链接。在磁盘中包含。4。关于生长功能结构的猜想。

项目成果

期刊论文数量（16）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

A search method for a thin position of a link

一种链接细位置的搜索方法

DOI：
发表时间：
2005
期刊：
Algebr. Geom. Topol. 5
影响因子：
0
作者：
T.Kobayashi;Y.Riek;Yasushi Yamashita;Yamashita Yasushi;M.Brittenham;Tsuyoshi Kobayashi
通讯作者：
Tsuyoshi Kobayashi

Essential laminations and branched surfaces in the exteriors of links

链节外部的基本叠片和分支表面

DOI：
发表时间：
2005
期刊：
Japan.J.Math 31
影响因子：
0
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

DOI：
发表时间：
期刊：
Journal fur die Reine and Ang. Math. (to appear)
影响因子：
0
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

Heegaard genus of the connected sum of m-small knots