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Deep learning the hyperbolic volume of a knot. (English) Zbl 1430.57001
Summary: 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, \(\operatorname{Vol}(K)\). A less studied question is whether \(\operatorname{Vol}(K)\) can be recovered directly from the original Jones polynomial \((N = 2)\). In this report, we use a deep neural network to approximate \(\operatorname{Vol}(K)\) 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.
MSC:
57-08 Computational methods for problems pertaining to manifolds and cell complexes
57K14 Knot polynomials
68T07 Artificial neural networks and deep learning
Software:
Mathematica; SnapPy
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