2006 Fiscal Year Final Research Report Summary
Invariants of knots and 3-manifolds
| Project/Area Number |
17540073
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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 | KYOTO UNIVERSITY |
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Principal Investigator |
OHTSUKI Tomotada Kyoto University, Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (50223871)
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| Co-Investigator(Kenkyū-buntansha) |
HABIRO Kazuo Kyoto University, Research Institute for Mathematical Sciences, Lecturer, 数理解析研究所, 講師 (80346064)
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| Project Period (FY) |
2005 – 2006
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| Keywords | knot / 3-manifold / invariant |
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| Research Abstract |
I organized researches on invariants of knots and 3-manifolds. I presented the 2-loop polynomial of a knot by using finite type invariants of degree * 4 of a spine of a Seifert surface of the knot, from the viewpoint that I regard the 2-loop polynomial of a knot as an "equivariant Casson invariant" of the infinite cyclic cover of the complement of the knot. Further, I studied equivariant quantum invariants of the infinite cyclic cover of the complement of a knot, which unify quantum invariants of finite covers branched along the knot. Further, I constructed invariants of knots from equivariant linking matrices of their surgery presentations, from the viewpoint of studying equivariant quantum U(1) invariant of the infinite cyclic covers of the complements of the knots. Furthermore, I formulated a perturbative invariant of a 3-manifold with the first Betti number 1 in a certain case, as the arithmetic perturbative expansion of quantum invariants of the 3-manifold.
I organized a low-dimensional topology seminar, jointly with Kazuo Habiro, who is a co-investigator of this research. The speakers were Yoko Mizuma, Takahito Kuriya, Soeren Hansen, Alexander Stoimenow, Vladimir Turaev, Alexis Virelizier, Gwenael Massuyeau, Masamichi Takase, and their talks were on advanced topics in the area of invariants of knots and 3-manifolds. I think their talks were very good, from the viewpoint of joint researches between them and me and the co-investigator, and from the viewpoint of research interactions between them and young researchers such as graduate students.
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