The generalized volume conjecture for the figure-eight knot parametrized by a complex number with small imaginary part

TL;DR

This work analyzes the large-$N$ asymptotics of the $N$-dimensional colored Jones polynomial for the figure-eight knot evaluated at $q=e^{\xi/N}$ with $0<\mathrm{Im}\xi<\pi/2$. Employing a Habiro-type summation, Poisson summation, and a saddle-point analysis, it relates the leading growth to the complex Chern-Simons data through $e^{N S(\xi)/\xi}$ and encodes subdominant behavior via the Alexander polynomial, $\Delta(e^{\xi})$. The paper introduces a detailed geometric framework of regions and curves (notably $\Gamma_{+}$, $\Gamma_{0}$, $\Gamma_{-}$) governing the asymptotics and proves precise formulas in each regime, including a $1/\Delta(e^{\xi})$ correction where the saddle contribution is absent or suppressed. A topological interpretation is provided by constructing non-Abelian PSL$(2,\mathbb{C})$ representations $\rho^{\pm}_{\xi}$, computing the adjoint Reidemeister torsion $T(\xi)$ and Chern–Simons invariants, and linking these invariants to the analytic data via $S(\xi)$ and its derivatives. The results extend the volume conjecture to a generalized setting with a complex parameter, bridging quantum invariants and hyperbolic geometry through explicit torsion and Chern–Simons calculations.

Abstract

We study the asymptotic behavior, as $N$ tends to infinity, of the $N$-dimensional colored Jones polynomial of the figure-eight knot, evaluated at $\exp(ξ/N)$ for a complex parameter $ξ$ with $0<\mathrm{Im}ξ<π/2$. We prove that if $\mathrm{Re}ξ$ is large the colored Jones polynomial grows exponentially with growth rate expressed by the Chern--Simons invariant, and that if $\mathrm{Re}ξ$ is small it converges to the reciprocal of the Alexander polynomial evaluated at $\expξ$.

Figures (13)

• Figure 1: The green lines correspond to Case (i), the yellow disks to (ii), the red lines to (iii) and (vi), the cyan striped region to (iv), the yellow diamond to (v), and the green disk to (vii).
• Figure 2: The region $\Xi$ is partitioned into the five regions $\Gamma_{+}$ (red), $\Gamma_{-}$ (cyan), $\widetilde{\Gamma}_{+}$ (pink), $\widetilde{\Gamma}_{-}$ (light cyan), and $\Omega\cap\Xi$ (striped cyan). The blue arc and the light blue arc indicate $\Gamma_{0}$ and $\widetilde{\Gamma}_{0}$, respectively.
• Figure 3: The red lines and the red region correspond to the representation $\rho^{+}_{\xi}$, and the green lines and the green disk to $\rho^{-}_{\xi}$, where $\rho^{+}_{\xi}$ ($\rho^{-}_{\xi}$, respectively) is a non-Abelian representation defining a (possibly incomplete) hyperbolic metric with positive (negative, respectively) volume. The yellow disks and the yellow diamond correspond to the affine representation $\rho^{\pm}_{\kappa}$. The cyan region and the cyan striped region correspond to the Abelian representation $\rho^{\rm{Abel}}_{\xi}$.
• Figure 4: $\Sigma$ (left) and $\Sigma_{\nu}$ (right).
• Figure 5: The function $f_N(z)$ is defined in the region $\Theta$, which is between the two red lines except for the four yellow triangles.

Theorems & Definitions (134)

• Conjecture 1.1: Volume conjecture Kashaev:LETMP97,Murakami/Murakami:ACTAM12001
• Conjecture 1.2: Complexification of the volume conjecture Murakami/Murakami/Okamoto/Takata/Yokota:EXPMA02
• Remark 1.3
• Remark 1.4
• Theorem 1.5: Main Theorem
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Remark 1.9
• Lemma 2.1