1998 Fiscal Year Final Research Report Summary
Applications of Seiberg-Witten theory to knot theory
| 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 Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (70160190)
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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)
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| Project Period (FY) |
1997 – 1998
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| Keywords | knot / link / three-manifold / quantum invariant / Vassiliev invariant / finite type invariant / Casson-Walker invariant / knot cobordism |
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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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