Elucidation of the geometric meaning of surface knot invariants defined by quandles and its application
| Project/Area Number |
17K05242
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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 | Tokyo Gakugei University |
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Principal Investigator |
TANAKA Kokoro 東京学芸大学, 教育学部, 准教授 (70448950)
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| Project Period (FY) |
2017-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| Keywords | 低次元トポロジー / 曲面結び目 / カンドル / 結び目 / トポロジー / bridge trisection / bridge trisection表示 / バイカンドル |
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| Outline of Final Research Achievements |
Surface knots were studied using quandle algebras and the following main results were obtained. (1) We explored a diagrammatic relation between biquandle colorings and quandle colorings, and showed that various invariants including coloring numbers are equivalent. (2) We gave a lower bound for the bridge indices of surface knots by using kei coloring numbers, and determined the explicit values for some specific examples. (3) We introduced the shifting chain map on quandle homology theory and explored a relation between quandle cocycle invariants and shadow cocycle invariants. We also observed the induced homomorphism. (4) We studied the knot quandles of knotted spheres, called twist-spun knots, and gave a example of knotted spheres with the same knot group but different knot quandles.
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| Academic Significance and Societal Importance of the Research Achievements |
古典的結び目は「ひもの結ばり方」を数学的に捉えた研究対象であり、曲面結び目は古典的結び目が「時間発展したもの」とみなすことができる。この曲面結び目という研究対象をカンドルと呼ばれる代数系を用いて調べることは重要である。今回得られた結果(1)と(3)は、カンドルの代数的な性質解明に新しい研究手段を与えるものである。また、今回得られた結果(2)と(4)は、曲面結び目の性質解明に新しい研究手段を与えるものである。なお、結果(4)は曲面結び目の結び目カンドルの代数的な性質に着目しており、今後の新たな研究への糸口となり得るものである。
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Report
(7 results)
Research Products
(15 results)
