2006 Fiscal Year Final Research Report Summary
Various inbariants appearing in low-dimensional topology
Grant-in-Aid for Scientific Research (B)
|Allocation Type||Single-year Grants |
|Research Institution||Tokyo Institute of Technorogy |
MURAKAMI Hitoshi Tokyo Institute of Technorogy, Department of Mathematics, Associate Professor -> 東京工業大学, 大学院理工学研究科, 助教授 (70192771)
TERASHIMA Yuji Tokyo Insitute of Technology, Department of Mathematics, Assistant Professor, 大学院理工学研究科, 助手 (70361764)
ISHIKAWA Masaharu Tokyo Insitute of Technology, Department of Mathematics, Assistant Professor, 大学院理工学研究科, 助手 (10361784)
KITANO Teruaki Soka University, Department of Information Systems Sciences, Associate Professor, 工学部, 助教授 (90272658)
USHIJIMA Akira Kanazawa University, Department of Mathematics, Lecturer, 大学院自然科学研究科, 講師 (50323803)
ENDO Hisaaki Osaka University, Department of Mathematics, Associate Professor, 大学院理学研究科, 助教授 (20323777)
|Project Period (FY)
2003 – 2006
|Keywords||knot / 3-mainfold / colored Jones polynomial / volume conjecture / quantum invariant / Chern-Simons invariant / Reidemeister torsion|
I am working on the volume conjecture for knots and its various generalizations. The volume conjecture states that a certain limit of the colored Jones polynomial of a knot would determine the volume of the complement of the knot. More precisely, we replace the paramaeter of the colored Jones polynomial of 'color' N with exp(2Pi^*I/V) and then consider the limit where N goes to the infinity. So far I generalized the volume conjecture as follows : If we replace the parameter with exp(a/N), change a variously, and take the limit, then we would have not only the volume of the knot complement but also the volume and the Chern-Simons invariant of the three-manifold obtained from the knot by Dehn surgery.
In this academic year, I studied not only the limit but also several coefficients of the asymptotic expansion of the logarithm of the colored Jones polynomial with respect to large N. The volume conjecture is equivalent to saying that the coefficient of N would determine the volume and the Chern-Simons invariant.
As a result of a joint work with S.Gukuv, we proposed the following new conjecture :
1.The coefficient of log N would be determined by the dimension of the cohomoloty group twisted by a representation of the knot group at SL(2, C).
2.The constant term would be determined by the Reidemeister torsion corresponding to a representation of the knot group at SL(2, C).
The conjecture above gives a new aspect to the volume conjecture and its generalizations. Moreover, we have confirmed by computer caluculations that this conjecture is true for some knots.
Research Products (13 results)