2002 Fiscal Year Final Research Report Summary
THE VOLUME CONJECTURE OF KNOTS AND ITS RAMIFICATIONS
Project/Area Number 
13640086

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  Tokyo Metropolitan University 
Principal Investigator 
YOKOTA Yoshiyuki Tokyo Metro. Univ., Grad. School of Sci., Assist. Prof., 理学研究科, 助教授 (40240197)

CoInvestigator(Kenkyūbuntansha) 
IMAI Jun Tokyo Metro. Univ., Grad. School of Sci., Assist. Prof., 理学研究科, 助教授 (70221132)

Project Period (FY) 
2001 – 2002

Keywords  colored Jones polynomial / volume conjecture / deformation space / Apolynomial 
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 ChernSimons 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 NeumanZaglerYoshida function and the Apolynomial. 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 NeumanZaglerYoshida function and whose partial differential equations gives the Apolynomial. The NeumanZaglerYoshida function is defined over deformation space of the hyperbolic structures of cusped hyperbolic manifolds and describes the analytic relation between the volumes and the ChernSimons invariants. The Apolynomial 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 Apolynomial 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.

Research Products
(8 results)