Colored Jones Polynomials and the Volume Conjecture

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.

@article{hughes2025_2502.18575,
  title={ Colored Jones Polynomials and the Volume Conjecture },
  author={ Mark Hughes and Vishnu Jejjala and P. Ramadevi and Pratik Roy and Vivek Kumar Singh },
  journal={arXiv preprint arXiv:2502.18575},
  year={ 2025 }
}