Disentangling a deep learned volume formula
Abstract
We present a simple phenomenological formula which approximates the hyperbolic volume of a knot using only a single evaluation of its Jones polynomial at a root of unity. The average error is just 2.86% on the first 1.7 million knots, which represents a large improvement over previous formulas of this kind. To find the approximation formula, we use layerwise relevance propagation to reverse engineer a black box neural network which achieves a similar average error for the same approximation task when trained on 10% of the total dataset. The particular roots of unity which appear in our analysis cannot be written as e^{2πi/(k+2)} with integer k; therefore, the relevant Jones polynomial evaluations are not given by unknotnormalized expectation values of Wilson loop operators in conventional SU(2) ChernSimons theory with level k. Instead, they correspond to an analytic continuation of such expectation values to fractional level. We briefly review the continuation procedure and comment on the presence of certain Lefschetz thimbles, to which our approximation formula is sensitive, in the analytically continued ChernSimons integration cycle.
 Publication:

Journal of High Energy Physics
 Pub Date:
 June 2021
 DOI:
 10.1007/JHEP06(2021)040
 arXiv:
 arXiv:2012.03955
 Bibcode:
 2021JHEP...06..040C
 Keywords:

 ChernSimons Theories;
 Topological Field Theories;
 Wilson;
 't Hooft and Polyakov loops;
 High Energy Physics  Theory;
 Computer Science  Machine Learning;
 Mathematics  Geometric Topology
 EPrint:
 v1: 26 + 19 pages, 15 figures