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

TL;DR

The paper constructs a quantum invariant $Z_gamma(M_u)$ for cusped hyperbolic 3-manifolds by assigning Faddeev's quantum dilogarithm to oriented ideal tetrahedra within a triangulation. In the classical limit, the integral reduces to the Neumann-Zagier potential, whose saddle points reproduce hyperbolic-gluing equations and allow extraction of the A-polynomial, linking quantum invariants to classical hyperbolic geometry and Chern-Simons data. This framework recovers known A-polynomials for knot complements and extends to once-punctured torus bundles, offering a non-compact generalization of the Jones–Witten invariant and a concrete realization of a generalized volume conjecture. The results suggest deep connections between quantum topology, hyperbolic geometry, Dehn surgery, and Bloch-type regulators, with potential computational and conceptual bridges to Mahler measures and Kasteleyn-type structures.

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.

Figures (8)

• Figure 1: Triangle with vertices $0$, $1$, and $z$ in $\mathbb{C}$. Here we set $z[1] = z$, $z[2] = 1 - \frac{1}{z}$, and $z[3] = \frac{1}{1-z}$.
• Figure 2: Pentagon identity \ref{['pentagon_S']} is interpreted as the Pachner move.
• Figure 3: Developing map of the complement of the figure-eight knot. The gray filled triangle corresponds to top vertex of the tetrahedron (central vertex of circle in \ref{['S_tetrahedron']}) with modulus $w$ in projection of \ref{['S_tetrahedron']}. A curve denotes a meridian of cusp.
• Figure 4: Developing map of the complement of $5_2$. Gray filled triangle denotes top vertex of the tetrahedron with modulus $z_1$. Meridian is denoted by a gray curve.
• Figure 5: Developing map of complement of the Pretzel knot $(-2,3,7)$. Horosphere of the top vertex of the tetrahedron with modulus $z_1$ is depicted by gray triangle. Gray curve denotes a meridian.