Colored Jones Polynomials and the Volume Conjecture

TL;DR

This paper computes colored Jones polynomials for hyperbolic knots and uses machine learning and polynomial evaluations to accurately predict knot volumes, proposing an improved volume conjecture based on phase evaluations.

Contribution

It introduces a novel approach combining polynomial computations, neural networks, and phase analysis to predict hyperbolic knot volumes and refines the volume conjecture.

Findings

Neural network predicts knot volume with 99.34% accuracy.

Polynomial evaluation at specific phases predicts volume nearly as well as neural networks.

Proposes an improved volume conjecture based on phase analysis of colored Jones polynomials.

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 $q=e^{ 8 \pi i / 15 }$ 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.