Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant

Details Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant

2006-04-05

arxiv.org/abs/math/0604094v1

J.Geom.Phys.57:1895-1940,2007

Mathematics

Quantum Algebra

Scientific paper

10.1016/j.geomphys.2007.03.008

We study quantum invariant Z(M) for cusped hyperbolic 3-manifold M. We construct this invariant based on oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in studies of the modular double of the quantum group. Following Thurston and Neumann-Zagier, we deform a complete hyperbolic structure of M, and correspondingly we define quantum invariant Z(M_u). This quantum invariant is shown to give the Neumann--Zagier potential function in the classical limit, 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 punctured torus bundle over the circle.

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