Papers › Colored Jones Polynomials and the Volume Conjecture
Colored Jones Polynomials and the Volume Conjecture
Mark Hughes, Vishnu Jejjala, P. Ramadevi, Pratik Roy, Vivek Kumar Singh
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.
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