GEOMETRY OF 3-MANIFOLDS AND QUANTUM INVARIANTS
| Project/Area Number |
15540089
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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., GRADUATE SCHOOL OF SCIENCE, ASSIST. PROFESSOR, 理学研究科, 助教授 (40240197)
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| Co-Investigator(Kenkyū-buntansha) |
MURAKAMI Jun WESEDA UNIV., GRADUATE SCHOOL OF SCIENCE AND ENGINEERING, PROFESSOR, 理工学術院, 教授 (90157751)
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| Project Period (FY) |
2003 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2004: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2003: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | knot / Jones polynomial / volume conjecture / A-多項式 |
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| Research Abstract |
The volume conjecture of knots states that the asymptotic behavior of the colored Jones polynomial, a genaralization of the famous Jones polynomial, determines the simplicial volume of the knot complement. This conjecture was first proposed by R. Kashaev for hyperbolic knots, and generalized by H. Murakami and J. Murakami for general knots. This conjecture is further generalized to involve the Chern-Simons invariants through a computer experiment made by H. Murakami, J. Murakami, M. Okamoto, T. Takata and the author. Now, many geometers and topologists are interested in this problem. The purpose of this research is to investigate the relationship between the geometry of 3-manifolds and the quantum invariants, motivated by the volume conjecture which suggests a relationship between the geometry of knot complements and the colored Jones polynomial.
In 2003, with H. Murakami, we proved that, for the figure eight knot, certain limit of the colored Jones polynomial dominates not only the vol
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ume of the complement but also the volumes of the closed 3-manifolds obtained by Dehn surgeries, which corrects the conjecture proposed by S. Gukov, and we proposed a new genaralization of the volume conjecture, which also explains the unexpected relationship between the recursive formula of the colored Jones polynomial and the A-polynomial of knots. We reported this result in the international workshops held at Edinburgh and Geneva in 2003 together with a newest result concerning the relationship between quantum 6j-symbols and volumes of hyperbolic tetrahedra. The author further confirmed that the argument for the figure eight knot is also available for so-called twist knots, and reported this result in the international workshop held at Potsdam in 2004.
On the other hand, when the author visited the University of Geneva in 2004, R. Kashaev and the author proved that the colored Jones polynomial of knots can be expressed by simple integrals over higher dimensional tori by using quantum dilogarithm functions whose asymptotic behaviors are well-known. We consider this result is a big progress toward the solution of the volume conjecture, because we may estimate the asymptotic behavior of such integrals over tori, a well-known compact manifold, by using the saddle point method together with the Morse theoretic argument. We have already reported this result in the meeting of American Mathematical Society held at Atlanta in early 2005. Less
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Report
(3 results)
Research Products
(11 results)
