Colored Jones Polynomials and the Volume Conjecture

TL;DR

This work correlates adjoint (3-colored) Jones polynomials with hyperbolic knot volumes by computing $J_3(K;q)$ via a vertex-model braid approach for hyperbolic knots up to $15$ crossings and demonstrating that neural networks can predict knot volumes from these polynomials with high accuracy. It identifies an optimal evaluation phase $q$ (notably $q=e^{8\pi i/15}$) that matches the predictive power of the neural model and proposes a phase-based refinement for the general $n$-colored Jones volume conjecture, $q(n)=\exp\left(2\pi i\frac{n+1}{n(n+2)}\right)$. The paper provides extensive data and a statistical/analytic analysis of zeros, degrees, and evaluations of $J_2$ and $J_3$, and derives a symbolic volume formula from $J_3$ evaluations with high $R^2$. It further tests an improved conjecture on knots with known higher-colored polynomials, showing faster convergence to volume and suggesting a deeper phase-structure in the volume conjecture. Overall, the results reinforce a strong link between colored Jones invariants and hyperbolic geometry while proposing concrete phase choices that enhance convergence and predictive power.

Abstract

Using the vertex model approach for braid representations, we compute polynomials for spin-1 placed on hyperbolic knots up to 15 crossings. These polynomials are referred to as 3-colored Jones polynomials or adjoint Jones polynomials. Training a subset of the data using a fully connected feedforward neural network, we predict the volume of the knot complement of hyperbolic knots from the adjoint Jones polynomial or its evaluations with 99.34% accuracy. A function of the adjoint Jones polynomial evaluated at the phase $q=e^{ 8 πi / 15 }$ predicts the volume with nearly the same accuracy as the neural network. From an analysis of 2-colored and 3-colored Jones polynomials, we conjecture the best phase for $n$-colored Jones polynomials, and use this hypothesis to motivate an improved statement of the volume conjecture. This is tested for knots for which closed form expressions for the $n$-colored Jones polynomial are known, and we show improved convergence to the volume.

Figures (16)

• Figure 1: Mean volume as a function of crossing number.
• Figure 2: Zeros of $J_2$ polynomials in our dataset in the (upper-half) complex $q$-plane.
• Figure 3: Zeros of $J_3$ polynomials in our dataset in the (upper-half) complex $q$-plane. Inset: Zoom in of the plot to the $x$-axis interval $(0.8,1.2)$ and $y$-axis interval $(0.0,0.05)$.
• Figure 4: Correlations between degrees of polynomials. Darker shades on points mean higher frequency.
• Figure 5: Distribution of minimum (blue) and maximum (green) degrees of $J_2$ (left subfigure) and $J_3$ (right subfigure). The $y$-axis corresponds to the number of times the given degree occurs as the minimum or maximum in our dataset.