３次元多様体のｇｒａｐｈｉｃ理論の展開と応用

三维流形图论的发展与应用

• 批准号: 11J09524
• 负责人: 長郷(井戸) 絢子 (2014)
• 金额: $ 1.72万
• 依托单位: Nara Women's University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2011
• 资助国家: 日本
• 起止时间: 2011-04-01 至 2015-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11J09524/
• 关键词: ヘンペル距離; ヒーガード分解; 曲線複体; 橋分解; 3次元多様体; graphic

Hempel[1]によって導入されたヒーガード分解の“距離”の概念は，どれくらい効率よく３次元多様体の断面（ヒーガード曲面）が取れているかを測っていると共に，３次元多様体の複雑さも捉えていることが多くの研究者によって明らかにされており，この概念を利用することで３次元多様体に関する様々な問題が解決されている．また，ヒーガード分解の距離を定義する上で用いられた曲線複体の概念は，Masur-Minsky[3,4,5]によって大きく進展し，現在も数多くの研究者によって活発に研究されており，特にここ数年，それを用いた様々な応用がみられている．本年度は，奈良女子大学の小林毅教授と張娟姫氏との共同研究において，これまでに得られた結果[2]である「曲線複体上での測地線の構成方法」を用いて「ヒーガード分解の距離を実現するような曲線複体上の測地線で結ばれる２つの頂点が一意に定まるようなヒーガード分解が存在する」という結果を両氏との共同研究により得ることができた．参考文献[1] J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001), no. 3. [2] A. Ido, Y. Jang, T. Kobayashi, Heegaard splittings of distance exactly n, Algebr. Geom. Topol., Mathematical Sciences Publishers, 14, 1395-1411(2014). [3] H. Masur and Y. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math., 138(1):103-149, 1999. [4] H. Masur and Y. Minsky, Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal., 10(4):902-974, 2000. [5] H. Masur and Y. Minsky, Quasiconvexity in the curve complex, In the Tradition of Ahlfors and Bers, III (W. Abikoff and A. Haas, eds.), Contemporary Mathematics 355, Amer. Math. Soc. (2004), 309-320.

Hempel [1] in this paper, we can analyze the concept of distance from each other, and analyze the data of the cross-section of 3-dimensional multi-body (curved surface). The results show that the cross-section of 3-dimensional multi-body (curved surface) is similar to that of the third-dimensional multi-body. In this paper, the concept is used to solve the problem of three-dimensional multi-body problem, the concept of curve copy is used in the definition of the distance from the definition, the concept of curve copy is used in the definition, and the concept is greatly improved in Masur-Minsky [3-4-4-5]. Now many researchers have been working on it for several years. This year, Professor Yoshiki Kobayi of Nara Women's University, Professor Yoshiki Kobayashi, has worked together to study the problem. The results of the simulation results [2] the method of the formation of the ground line of the earth on the complex of the curve [2] the method of the formation of the ground line on the curve complex is not valid. The distance is decomposed by using the computer to analyze the distance. The results show that the earth survey is performed on the complex curve of the computer. Reference [1] J. Hempel 3-manifolds as viewed from the curve complex, Topology 40 (2001), no. 3. [2] A. Ido, Y. Jang, T. Kobayashi, Heegaard splittings of distance exactly n, Algebr. Geom. Topol., Mathematical Sciences Publishers, 14, 1395-1411 (2014). [3] H. Masur and Y. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math., 138 (1): 103-149, 1999. [4] H. Masur and Y. Minsky, Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal., 10 (4): 902-974, 2000. [5] H. Masur and Y. Minsky, Quasiconvexity in the curve complex, In the Tradition of Ahlfors and Bers, III (W. Abikoff and A. Haas, eds.), Contemporary Mathematics 355, Amer. Math. Soc. (2004), 309320

项目成果

On keen Heegaard splitting (joint work with Yeonhee Jang (Nara Women’s University) and Tsuyoshi Kobayashi(Nara Women’s University))

关于敏锐的 Heegaard 分裂（与 Yeonhee Jang（奈良女子大学）和 Tsuyoshi Kobayashi（奈良女子大学）合作）

• 发表时间: 2015
• 作者: A.Ido;Y.Jang and T.Kobayashi;Ayako Ido;Ayako Ido
• 通讯作者: Ayako Ido

結び目の橋分解の距離について

关于结桥分解距离

• 发表时间: 2012
• 作者: A.Ido;Y.Jang and T.Kobayashi;Ayako Ido;Ayako Ido;井戸 絢子;Ayako Ido;Ayako Ido;Ayako Ido;井戸 絢子
• 通讯作者: 井戸 絢子

Heegaard splitting with distance exactly $n$ for each non-negative integer $n$

对于每个非负整数 $n$，Heegaard 分裂的距离恰好为 $n$

• 发表时间: 2012
• 作者: A.Ido;Y.Jang and T.Kobayashi;Ayako Ido;Ayako Ido;井戸 絢子;Ayako Ido;Ayako Ido;Ayako Ido
• 通讯作者: Ayako Ido

On the distance of Heegaard splitting for a closed 3-manifold

关于封闭 3 流形的 Heegaard 分裂距离

• 发表时间: 2012
• 作者: A.Ido;Y.Jang and T.Kobayashi;Ayako Ido;Ayako Ido;井戸 絢子;Ayako Ido;Ayako Ido
• 通讯作者: Ayako Ido

On genus g Heegaard splittings with distance 2g

在距离为 2g 的 g Heegaard 分裂上

• 发表时间: 2011
• 作者: A.Ido;Y.Jang and T.Kobayashi;Ayako Ido;Ayako Ido;井戸 絢子;Ayako Ido;Ayako Ido;Ayako Ido;井戸 絢子;井戸 絢子;井戸絢子;井戸絢子;井戸絢子;井戸絢子;井戸絢子
• 通讯作者: 井戸絢子