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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
Full Text:
References:
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