TY - JOUR

T1 - Generalized volume conjecture and the A-polynomials

T2 - The Neumann-Zagier potential function as a classical limit of the partition function

AU - Hikami, Kazuhiro

N1 - Funding Information:
The author would like to thank H. Murakami, K. Shimokawa, and T. Takata for communications. He also thanks R. Benedetti for bringing Refs. [2,3] to attention. We have used the computer programs SnapPea [60] , Knotscape [28] , and Snap [14] , in studying triangulation of manifolds. We have also used Mathematica and Pari/GP. Pictures of knots in this paper are drawn using KnotPlot [54] . This work is supported in part by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

PY - 2007/8

Y1 - 2007/8

N2 - We introduce and study the partition function Zγ (M) for the cusped hyperbolic 3-manifold M. We construct formally this partition function based on an oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in his studies of the modular double of the quantum group. Following Thurston and Neumann-Zagier, we deform a complete hyperbolic structure of M, and we define the partition function Zγ (Mu) correspondingly. This function is shown to give the Neumann-Zagier potential function in the classical limit γ → 0, and the A-polynomial can be derived from the potential function. We explain our construction by taking examples of 3-manifolds such as complements of hyperbolic knots and a punctured torus bundle over the circle.

AB - We introduce and study the partition function Zγ (M) for the cusped hyperbolic 3-manifold M. We construct formally this partition function based on an oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in his studies of the modular double of the quantum group. Following Thurston and Neumann-Zagier, we deform a complete hyperbolic structure of M, and we define the partition function Zγ (Mu) correspondingly. This function is shown to give the Neumann-Zagier potential function in the classical limit γ → 0, and the A-polynomial can be derived from the potential function. We explain our construction by taking examples of 3-manifolds such as complements of hyperbolic knots and a punctured torus bundle over the circle.

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U2 - 10.1016/j.geomphys.2007.03.008

DO - 10.1016/j.geomphys.2007.03.008

M3 - Article

AN - SCOPUS:34248578237

SN - 0393-0440

VL - 57

SP - 1895

EP - 1940

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

IS - 9

ER -