2000 Fiscal Year Final Research Report Summary
Topological field theory and some problems on 3-manifolds
| Project/Area Number |
11640085
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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 | TOKYO METROPOLITAN UNIVERSITY (2000)
Kyushu University (1999) |
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Principal Investigator |
YOKOTA Yoshiyuki Tokyo Metropolitan University, Department of Mathematics, Assistant Professor, 理学研究科, 助教授 (40240197)
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| Project Period (FY) |
1999 – 2000
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| Keywords | knot / 3-manifold / quantum invariant / volume conjecture |
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| Research Abstract |
The purpose of this research is to attack various problems on knots and 3-manifolds by using quantum invariants deribed from topological quantum field theory, and we obtain a mysterious relationship, which is well-known as the volume conjecture of knots, between the geometric structure of knots and 3-manifolds and the asymptotic behavior of quantum invariants as described below.
In 1999, for a knot in 3-sphere, we first extracted a characteristic complex function from the integral approximation of the Jones invariant. Then, for a hyperbolic knot in 3-sphere, we found a surprising correspondence between the stationary phase equations for the function and the hyperbolicity equations deduced from an ideal triangulation of the complement. Furthermore, we observed that a critical value of the function gives the volume and the Chern-Simons invariant of the complement. Following these results, in 2000, we obtain a family of potential functions, which dominates the deformation space of hyperbolic structures of the complement, by adding some term coming from the triangulation to the function above. That is, the stationary phase equations for a potential function coincide with the hiperbolicity equations for a non-complete hyperbolic structure, and a critical value of the potential function gives not only the volume but also the Chern-Simons invariant of the closed, hyperbolic 3-manifold obtained from the complement by appropriate Dehn surgery.
Recently, some experiments reveal that the volumes of hyperbolic tetrahedra are related to quantum 6j-symbols, which are used to define the quantum invariants of closed 3-manifolds. Conbining these results with ours, we now have a strong motivation to generalize the volume conjecture to closed 3-manifolds.
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Research Products
(2 results)
