2006 Fiscal Year Final Research Report Summary
General research of the volume conjecture of knots
| Project/Area Number |
17540090
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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 Metropolitan University, Graduate School of Science and Engineering, Assistant Professor, 大学院理工学研究科, 准教授 (40240197)
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| Co-Investigator(Kenkyū-buntansha) |
MURAKAMI Jun Waseda University, Graduate School of Science and Engineering, Professor, 理工学術院, 教授 (90157751)
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| Project Period (FY) |
2005 – 2006
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| Keywords | knot / Jones polynomial / volume conjecture / Akutsu-Deguchi-Ohtsuki invariant |
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| Research Abstract |
The volume conjecture of knots states that the asymptotic behavior of the colored 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 was further generalized to involve the Chern-Simons invariant through computer experiments made by H.Murakami, J.Murakami, M.Okamoto, T.Takata and Y.Yokota. Finally, S.Gukov and ft Murakami conjectured that various limit of the colored Jones polynomial dominates not only the volume of the knot complement but also the volumes of the closed 3-manifolds obtained by Dehn surgeries, or the Neumann-Zagier function on the deformation space of the knot complement. Our strategy to prove the hyperbolic volume conjecture, the first one proposed by Kashaev, is :
(1) express the colored Jones polynomial as an integral over a torus
(2) apply the saddle point method by using Morse theory
Along this strategy, with Kashaev at University of Geneva, we achieved (1) for any knot, and (2) for some knots. We have reported these in many workshops abroad, and I am writing a paper on the Neumann-Zagier function of knot complements, which is necessary to connect the limit of the colored Jones polynomial with the volume, a paper to describe the algorithm to realize (1) for any knot and a paper on (2) for some knots simulteneoualy. On the other hand, the co-researcher J. Murakami observed that the asymptotic behavior of the Akutsu-Deguchi-Ohtsuki invariant of Whitehead link and Borromean rings gives the volumes of the orbifolds they define. This is an interesting generalization of the conjecture made by S.Gukov and H.Murakami.
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