2014 Fiscal Year Final Research Report
On the volume conjecture for knots
| Project/Area Number |
24540088
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Tokyo Metropolitan University |
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Principal Investigator |
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| Project Period (FY) |
2012-04-01 – 2015-03-31
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| Keywords | 結び目 / ジョーンズ多項式 / 体積予想 / ポテンシャル関数 / カスプ・シェイプ |
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| Outline of Final Research Achievements |
Volume conjecture for knots states that, for a knot in 3-sphere, the volume of its complement appears in the limit of its Jones polynomial. This is very important conjecture because the geometric back ground of quantum invariants such as Jones polynomials is not clear yet. To prove this conjecture, we have to study the geometric property of the potential function which appears in the integral expression of the Jones polynomial of the knot. In fact, it is already know that the stationary phase equations and a critical value of the potential function give the structure equations and the complex volume of the knot.
In this research, we prove that the Hessian of the potential function gives the cusp shape, a geometric invariant of knots. Furthermore, we give a formula for the generalized cusp shape in terms of the potential function.
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| Free Research Field |
位相幾何学
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