2017 Fiscal Year Final Research Report
On the volume conjecture for knots and potential functions
| Project/Area Number |
15K04878
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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) |
2015-04-01 – 2018-03-31
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| Keywords | 体積予想 / ポテンシャル関数 / 交代結び目 |
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| Outline of Final Research Achievements |
The volume conjecture for knots states that, for a knot in 3-sphere, the volume of its complement appears in the limit of its colored Jones polynomial. This is very important conjecture because the geometric background of quantum invariants, such as Jones polynomials, is still unclear. To prove this conjecture, we have to study the geometric and analytic properties of the potential function which appears in the integral expression of the colored Jones polynomial. In fact, it is already known that the stationary phase equations and the critical value of the potential function give the structure equations and the volume.
In this reaserch, we study the geodesics in the complements of the alternating knots, the existence of the solution to the structure equations, and a numerical method to compute the A-polynomial by using the derivatives of the potential function.
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| Free Research Field |
位相幾何学
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