| Project/Area Number |
09640135
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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 | Waseda University |
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Principal Investigator |
UENO Kimio (1998) Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (70160190)
村上 斎 (1997) 早稲田大学, 理工学部, 助教授 (70192771)
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| Co-Investigator(Kenkyū-buntansha) |
FUKUYAMA Masaru Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (80063741)
KOJIMA Jun Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (50063540)
村上 斉 , 助教授 (70192771)
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| Project Period (FY) |
1997 – 1998
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| Project Status |
Completed (Fiscal Year 1998)
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| Budget Amount *help |
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1998: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 1997: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Keywords | knot / link / three-manifold / quantum invariant / Vassiliev invariant / finite type invariant / Casson-Walker invariant / knot cobordism / 結び目 / 絡み目 / Vassiliew不変量 / 量子不変量 / Reidemeister torsion / Seiberg-Witten不変量 / Seiberg-Witten理論 / 結び目理論 / 4次元多様体 / 3次元多様体 |
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| Research Abstract |
I will describe my results paper by paper.
In the paper (i) I gave a new elementary, combinatorial definition of the HOMFLY polynomial. In (ii) I looked at the multivariable Alexander polynomial of links from the view point of Vassiliev invariants and define a recursive definition of weight systems derived from it. In (iii) we studied a knot cobordism invariant, 4-dimensional clasp number, introduced by T.Shibuya. He proved that it is greater than or equal to the 4-dimensional genus and raised a problem whether there are knots which do not satisfy the equality. We gave such an example in this paper. In (iv) I studied the quantum SU(2)-invariant of 3-manifolds associated with the gammath root of unity. If gamma is even, it is defined for a class of the first cohomology group modulo 2. In this paper I calculated it for rational homology three-spheres and for the trivial cohomology class and showed that it is a cyclotomic integer and moreover it determines the Casson-Walker invariant. In (v) we introduced a filtration to the vector space spanned by all the Seifert matrices corresponding to the filtration to the vector space spanned by all the knot, which was introduced by V.Vassiliev. Moreover we clarified its relation to the Alexander polynomial. In (vi) I gave an example of hyperbolic three-manifold with trivial finite type invariants (introduced by T.Ohtsuki) up to arbitrarily given degree. In (vii) I followed (iv) and obtained a similar result in the case of non-trivial cohomology classes. Unfortunately I only showed that the invariant is a cyclotomic integer and a relation to the Casson-Walker invariant is now being investigated.
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