Deep Learning the Hyperbolic Volume of a Knot

Abstract

An important conjecture in knot theory relates the large-$N$, double scaling limit of the colored Jones polynomial $J_{K,N}(q)$ of a knot $K$ to the hyperbolic volume of the knot complement, $\text{Vol}(K)$. A less studied question is whether $\text{Vol}(K)$ can be recovered directly from the original Jones polynomial ($N=2$). In this report we use a deep neural network to approximate $\text{Vol}(K)$ from the Jones polynomial. Our network is robust and correctly predicts the volume with $97.6\%$ accuracy when training on $10\%$ of the data. This points to the existence of a more direct connection between the hyperbolic volume and the Jones polynomial.

arXiv comment18 pages, 9 figures, updated figures

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• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
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• Machine Learning of Calabi-Yau Volumes
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