Deep learning the hyperbolic volume of a knot

• DOI: 10.1016/j.physletb.2019.135033
• Publication Date: 2019-10-28T16:27:39Z
• Authors: Vishnu Jejjala; Arjun Kar; Onkar Parrikar
• Keywords: High Energy Physics - Theory; Physics; QC1-999; FOS: Physical sciences; Geometric Topology (math.GT); 01 natural sciences; Mathematics - Geometric Topology; High Energy Physics - Theory (hep-th); Mathematics - Quantum Algebra; 0103 physical sciences; FOS: Mathematics; Quantum Algebra (math.QA); 0101 mathematics

18 pages, 9 figures, updated figures<br/>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.<br/>