Geometric structures of 3-manifolds and various related structures
| Project/Area Number |
17540077
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Nara Women's University |
|---|
Principal Investigator |
KOBAYASHI Tsuyoshi Nara Women's University, Graduate School of Humanities and Sciences, Professor, 大学院人間文化研究科, 教授 (00186751)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
YAMASHITA Yasushi Nara Women's University, Faculty of Science, Associate Professor, 理学部, 助教授 (70239987)
KATAGIRI Minnyou Nara Women's University, Faculty of Science, Associate Professor, 理学部, 助教授 (60263422)
|
|---|
| Project Period (FY) |
2005 – 2006
|
| Project Status |
Completed (Fiscal Year 2006)
|
| Budget Amount *help |
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
|
|---|
| Keywords | 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 |
|---|
| Research Abstract |
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 groups
We made computar experiments on the growth functions of 2-bridge link groups, and posed conjectures on the structure of the growth functions.
|
Report
(3 results)
Research Products
(16 results)
