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

Abstract

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.

References in corpus (3)

• Difference equation of the colored Jones polynomial for torus knot
• A dilogarithmic formula for the Cheeger-Chern-Simons class
• Some limits of the colored Jones polynomials of the figure-eight knot

Cited by in corpus (11)

• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• Quantum geometry of 3-dimensional lattices
• A short overview of the "Topological recursion"
• Chern-Simons Theory and S-duality
• The Volume Conjecture and Topological Strings
• The Hitchin-Witten Connection and Complex Quantum Chern-Simons Theory
• Complex Chern-Simons theory at level k via the 3d-3d correspondence
• A Spectral Perspective on Neumann-Zagier
• Braiding Operator via Quantum Cluster Algebra
• Asymptotic aspects of the Teichmüller TQFT