2002 Fiscal Year Final Research Report Summary
THE VOLUME CONJECTURE OF KNOTS AND ITS RAMIFICATIONS
| Project/Area Number |
13640086
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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 |
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Principal Investigator |
YOKOTA Yoshiyuki Tokyo Metro. Univ., Grad. School of Sci., Assist. Prof., 理学研究科, 助教授 (40240197)
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| Co-Investigator(Kenkyū-buntansha) |
IMAI Jun Tokyo Metro. Univ., Grad. School of Sci., Assist. Prof., 理学研究科, 助教授 (70221132)
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| Project Period (FY) |
2001 – 2002
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| Keywords | colored Jones polynomial / volume conjecture / deformation space / A-polynomial |
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| Research Abstract |
The volume conjecture of knots states that the asymptotic behavior of the colored Jones polynomials, a generalization of the famous Jones polynomial, determines the simplicial volume of the complement of a knot, which is proposed by Kashaev, J. Murakami and H. Murakami. This conjecture is now generalized to involve the Chern-Simons invariant through the computer experiment by H. Murakami, J. Murakami, M. Okamoto, T. Takata and the author, and many mathematicians are now interested in this problem.
The purpose of this research is to investigate the relationship between the geometry of the knot complement and the colored Jones polynomials, and we have shown that the colored Jones polynomials dominate both the Neuman-Zagler-Yoshida function and the A-polynomial. In fact, from the study of the volume conjecture, we derive a potential function from the colored Jones polynomials which can be deformed into the Neuman-Zagler-Yoshida function and whose partial differential equations gives the A-polynomial.
The Neuman-Zagler-Yoshida function is defined over deformation space of the hyperbolic structures of cusped hyperbolic manifolds and describes the analytic relation between the volumes and the Chern-Simons invariants. The A-polynomial of knots is defined from the representation space of the fundamental groups of knots and plays an important role in the theory of Dehn surgery. Recently, some number theorists are also interested in the A-polynomial because its Mahler measure gives certain Dedekind zeta function.
The author was invited to many conferences and had many opportunities to exhibit our results. The author hopes many mathematicians are interested in this research.
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Research Products
(8 results)
