2002 Fiscal Year Final Research Report Summary
Topological invariants of manifolds and modular forms
| Project/Area Number |
12640089
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Tsuda College |
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Principal Investigator |
FUKUHARA Shinji Tsuda College, Faculty of Liberal Arts, Professor, 学芸学部, 教授 (20011687)
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| Co-Investigator(Kenkyū-buntansha) |
MIYAZAWA Aida Haruko Tsuda College, Institute of Math. Comp. Sci., Researcher, 数学計算機科学研究所, 研究員 (40266276)
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| Project Period (FY) |
2000 – 2002
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| Keywords | manifold / topological invariant / modular form / Dedekind sum / knot / link / elliptic function |
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| Research Abstract |
The head investigator has been studying the number theoretical functions which appear to be topological invariants of manifolds and knots. In particular he has been studying Dedekind symbols. His research is based on the fact that Dedekind symbols with Laurent polynomial reciprocity law correspond to modular forms bijectively. His first result is published in the article "Twisted generalized Dedekind symbols, Shinji Fukuhara, J. Nunber Theory 82 (2000) 47-78" which developed the theory of twisted Dedekind symbols.
The second result is a joint work which was published in "Non-commutative polynomial reciprocity law, Shinji Fukuhara, Yukio Matsumoto, Noriko Yui, Internat. J. Math. 12 (2001) 973-986". Here the authors demonstrate that topology of a torus is closely related to reciprocity law.
The third result is finding the method of obtaining Dedekind symbols from Jacobi forms. It was published in "Dedekind symbols associated with J-forms and their reciprocity law, Shinji Pukuhara, J. Number Theory 98 (2003) 236-253". Futher- ***re he discovered the explicit formulae for two-bridge knot polynomials. The formulae are written in terms of new Dedekind symbols. The result will be published in J. Austral. Math. Soc.
The other investigator of this project also obtained the result on local moves and Vassiliev invariants which is prepared to be published with the title "SCn-moves and (n + l)-st coefficients of the Conway polynomials of links, Haruko Aida Miyazawa".
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Research Products
(12 results)
