Modular forms and Dedekind symbols as topological invariants
| Project/Area Number |
16540081
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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 Haruko Tsuda College, Institute of Math. and Comp. Sci., Research Fellow, 数学・計算機科学研究所, 研究員 (40266276)
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| Project Period (FY) |
2004 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
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| Keywords | topological invariant / knot / manifold / Dedekind sum / modular form / Hecke operator / period polynomial / 尖点形式 / 周期積分 / 絡み目 |
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| Research Abstract |
The head investigator has been studying relationship between Dedekind symbols and knot (manifold) invariants. For example, he showed that Conway polynomials of two-bridge knot are given using Dedekind symbols in his paper "Explicit formulae for two-bridge knot polynomials, J. Aust. Math. Soc. 78 (2005), 149-166". Dedekind symbols with polynomial reciprocity laws are especially important. He found explicit formulas for Dedekind symbols with polynomial reciprocity laws and presented the result in his paper "Dedekind symbols with reciprocity laws, Math. Ann. 329 (2004), 315-334".
It is known that there is natural correspondences between Dedekind symbols, modular forms and period polynomials. He found that Hecke operators on Dedekind symbols can be defined so that they are compatible with Hecke operators on modular forms and period polynomials. As a corollary, he obtained explicit formulas for Hecke matrices of cusp forms. The results are published in the papers "Hecke operators on weighted Dedekind symbols, J. reine angew. Math. 593 (2006) 1-29" and "Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials, J. reine angew. Math. (in print) ".
The investigator Miyazawa studied how knot invariants change when knots are moved locally. She gave a talk on this subject at the meeting for "knots and low dimensional manifolds" and published a joint paper "Classification of n-component Brunnian links up to C_n-move, Topology Appl. 153 (2006) 1643-1650" with Akira Yasuhara.
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Report
(4 results)
Research Products
(17 results)
