Rationality of the parametrized volume conjecture coefficients and the Reidemeister torsion normalization for hyperbolic knots

Open problems posed by Dubois and Garoufalidis in 'Rationality of the SL(2,C)-Reidemeister torsion in dimension 3' (arXiv:0908.1690; Topology Proc. 47 (2016) 115-134). The source proves that the non-abelian Reidemeister torsion of a hyperbolic 3-manifold with torus boundary is a rational function germ on the SL(2,C) character variety and poses four follow-up questions. The objective is two of the source's follow-up questions: (Problem 1.7) for the asymptotic coefficients $c_k(\alpha)$ of the colored Jones polynomial $J_{K,N}(e^{\alpha/N})$ in the parametrized volume conjecture regime, prove that for every $k$ each coefficient, suitably normalized, is the germ of a unique element of the rational function field $\mathbb{Q}(X_M)$ of the character variety; and (Conjecture 1.8) prove the leading-term normalization $c_0(0) = (2\tau_\mu(\rho_0))^{-1/2}$ identifying the colored Jones prefactor with the Reidemeister torsion at the discrete faithful representation. Both are established only for the figure-eight knot, twice-iterated torus knots, and two-bridge knots; the general hyperbolic case is open. The source's remaining follow-up questions are outside this objective: Problem 1.9 (higher-rank torsion rationality) is substantially resolved by later literature, and Problem 1.5 (nontriviality of the $T_\lambda$-polynomial) is informally posed.

Background

A hyperbolic knot $K \subset S^3$ is a knot whose complement $M = S^3 \setminus K$ carries a complete finite-volume hyperbolic metric; equivalently, the interior of $M$ is hyperbolic 3-space quotiented by a discrete faithful representation $\rho_0$, which takes values in $\mathrm{PSL}(2,\mathbb{C})$ (the orientation-preserving isometry group of hyperbolic 3-space) and lifts to $\mathrm{SL}(2,\mathbb{C})$. The $\mathrm{SL}(2,\mathbb{C})$ character variety $X_M$ is the algebraic variety whose points record the conjugacy classes of representations of the knot group $\pi_1(M)$ into $\mathrm{SL}(2,\mathbb{C})$ via their traces; near $\rho_0$ it is a smooth complex curve, and $\mathbb{Q}(X_M)$ denotes its field of rational functions with rational coefficients. The $A$-polynomial of $K$ is the two-variable polynomial cutting out the closure of the image of $X_M$ in the plane of eigenvalues $(e_\mu, e_\lambda)$ of the meridian $\mu$ and longitude $\lambda$ of the knot. The non-abelian (twisted) Reidemeister torsion $\tau_\gamma$ associated to a peripheral curve $\gamma$ is an analytic function on $X_M$ near $\rho_0$ built from the Reidemeister torsion of the cochain complex of $\pi_1(M)$ twisted by a representation, made well defined near $\rho_0$ after fixing conventions; Porti's memoir established its analyticity and local behavior. Dubois and Garoufalidis proved (their Theorem 1.2, following Porti) that $\tau_\gamma$ is the germ of an element of $\mathbb{Q}(X_M)$, and (their Theorem 1.4) that for every admissible curve $\gamma$ there exists a two-variable integer polynomial $T_\gamma(\tau, y)$, the $T_\gamma$-polynomial, with $T_\gamma(\tau_\gamma, \mathrm{tr}_\gamma) = 0$ identically near $\rho_0$.

The colored Jones polynomial $J_{K,N}(q)$ is the quantum-group invariant of $K$ colored by the $N$-dimensional irreducible representation of $\mathfrak{sl}_2(\mathbb{C})$, normalized to be $1$ at the unknot. The volume conjecture asserts that $J_{K,N}$ evaluated at the root $q = e^{2\pi i/N}$ grows exponentially at a rate given by the hyperbolic volume of $M$. The parametrized volume conjecture of Gukov-Murakami and Garoufalidis-Le refines this by moving the evaluation point: for fixed complex $\alpha$ near $2\pi i$ one studies the sequence $J_{K,N}(e^{\alpha/N})$, which is expected to admit an asymptotic expansion

$J_{K,N}(e^{\alpha/N}) \sim e^{N\,\mathrm{CS}(\rho_\alpha)/(2\pi i)}\,N^{3/2}\,c_0(\alpha)\left(1 + \sum_{k=1}^{\infty}\frac{c_k(\alpha)}{N^k}\right),$

where $\rho_\alpha$ is the representation near $\rho_0$ with $\mathrm{tr}_\mu(\rho_\alpha) = e^\alpha + e^{-\alpha}$ and $\mathrm{CS}$ is the $\mathrm{SL}(2,\mathbb{C})$ Chern-Simons invariant, a complex-valued refinement of the hyperbolic volume. Perturbative Chern-Simons theory with complex gauge group predicts that the 0-loop term of such an expansion is the classical action, the 1-loop term is the Reidemeister torsion, and the higher-loop terms are computable corrections; Dimofte-Gukov-Lenells-Zagier computed the exact perturbative expansion for the figure-eight knot, and Murakami extracted from it the Chern-Simons invariant and the torsion. The source paper asks whether this entire expansion is governed by the function field of the character variety in the same way the torsion itself is: it conjectures that all perturbative coefficients $c_k(\alpha)$ are germs of elements of $\mathbb{Q}(X_M)$ and that the leading coefficient is exactly the torsion prefactor $(2\tau_\mu(\rho_0))^{-1/2}$.

Problem Statement

Let $K \subset S^3$ be a hyperbolic knot with complement $M = S^3 \setminus K$, discrete faithful representation $\rho_0$ lifted to $\mathrm{SL}(2,\mathbb{C})$, character variety $X_M$, and rational function field $\mathbb{Q}(X_M)$; let $\mu$ denote a meridian of $K$. In the parametrized volume conjecture regime, for fixed complex $\alpha$ near $2\pi i$ the colored Jones polynomial is expected to satisfy

$J_{K,N}(e^{\alpha/N}) \sim e^{N\,\mathrm{CS}(\rho_\alpha)/(2\pi i)}\,N^{3/2}\,c_0(\alpha)\left(1 + \sum_{k=1}^{\infty}\frac{c_k(\alpha)}{N^k}\right),$

where $\rho_\alpha$ is the branch of representations near $\rho_0$ with $\mathrm{tr}_\mu(\rho_\alpha) = e^\alpha + e^{-\alpha}$ and $\mathrm{CS}$ is the $\mathrm{SL}(2,\mathbb{C})$ Chern-Simons invariant. Because the peripheral holonomy of the complete hyperbolic structure is parabolic, the lifted discrete faithful representation satisfies $\mathrm{tr}_\mu(\rho_0) = \pm 2$; since $e^\alpha$ is $2\pi i$-periodic, the branch $\rho_\alpha$ satisfies $\rho_{\alpha=0} = \rho_{\alpha=2\pi i} = \rho_0$, the complete hyperbolic structure, and $\alpha$ near $2\pi i$ places $\rho_\alpha$ near $\rho_0$. Accordingly, $c_0(0)$ in part (ii) denotes the value at $\rho_0$ of the germ of $c_0$ viewed (per part (i)) as an element of $\mathbb{Q}(X_M)$, equivalently the limit of $c_0(\alpha)$ as $\alpha \to 2\pi i$ within the expansion domain. This matches the shifted convention $q = e^{(u+2\pi i)/N}$ with $u = 0 \leftrightarrow \rho_0$ (meridian eigenvalues $e^{\pm u/2}$) used in the verified cases: Murakami for the figure-eight knot, and Ohtsuki-Takata for two-bridge knots with the torsion taken at the holonomy representation of the complete hyperbolic structure; these verified cases concern the same hyperbolic evaluation point $\rho_0$ and the same $\mathbb{Q}(X_M)$-membership content as (i)-(ii). Prove, for every hyperbolic knot $K$:

(i) (Problem 1.7 of the source.) For every $k \geq 0$, with suitable normalization, $c_k(\alpha)$ is the germ of a unique element of the field $\mathbb{Q}(X_M)$, i.e. every perturbative coefficient of the colored Jones asymptotics is a rational function on the character variety with rational coefficients.

(ii) (Conjecture 1.8 of the source.) The leading coefficient satisfies the torsion normalization at the hyperbolic point

$c_0(0) = \left(2\,\tau_\mu(\rho_0)\right)^{-1/2},$

where $\tau_\mu$ is the non-abelian Reidemeister torsion twisted by the meridian, evaluated at the discrete faithful representation.

These two items form a single scientific objective: to place the complete perturbative semiclassical expansion of the colored Jones polynomial of a hyperbolic knot into the same function field that governs the traces and the torsion, with the torsion fixing the leading normalization. Part (ii) is the $k = 0$, $\alpha = 0$ specialization of part (i) together with the precise identification of the prefactor, so an answer to (i) naturally carries (ii) as its first consistency check. The claim is over all hyperbolic knots; the verified cases (figure-eight, twice-iterated torus knots, two-bridge knots) do not settle it, and neither does the equivalent 1-loop statement known for fibered manifolds and fundamental shadow links. The source's Problem 1.9 is not part of this objective, since later literature substantially resolved it, and neither is Problem 1.5, whose literal reading is settled trivially by the source's own Theorem 1.4 while its intended reading lacks a precise formulation.

References

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