2020 Fiscal Year Final Research Report
Development and Evolution of Low-dimensional Topology
| Project/Area Number |
17K05249
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Nara Women's University |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2021-03-31
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| Keywords | Heegaard 分解 / 三次元多様体 / 結び目 / 絡み目 / 橋分解 / 折り紙 / データ解析 |
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| Outline of Final Research Achievements |
For Purpose 1), we gave a necessary and sufficient condition for existing keen Heegaard splitting. For Purpose 2), we studied the structure of the complex obtained from the set of the spherical curves. We gave several results that give estimations of simplicial distances between vertices. Particularly we could give a complete description of the subcomplex consisting of the vertices arising from the spherical curves with at most eight double points. We defined a new invariant called “stable double point number” for a pair of spherical curves, and by making use of it we showed that there is a spherical curve that is transformed to a trivial spherical curve by a sequence of RI or RIII moves such that the number of the double points has to be raised by 2 during the process of the deformation.
For Purpose 3), we proposed a method for producing flat foldable origamis. We studied about neural networks for targeting to originating new research field in topology.
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| Free Research Field |
幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
この研究によりHeegaard分解に関する新しい視点が導入された．その成果はもちろんのこと，研究にあたって，様々な具体例をつくるのに必要な技巧を提供でき，これからの大きな成果が期待できる．結び目・絡み目の橋表示という，古典的な研究対象にその基本的な部分で研究すべき領域が明らかになった．これからの研究を進める上での方向性を明らかにすることができた．低次元トポロジーに関してはパーシステントホモロジーの理論等実生活に結びついた応用が発見されている．本研究で低次元トポロジーの様々な成果を実用的に結び付けることを意識した研究ができた．特にその展開の可能性は大きいと考える．
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