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 layer-wise 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 e2πi/(k+2) with integer k; therefore, the relevant Jones polynomial evaluations are not given by unknot-normalized expectation values of Wilson loop operators in conventional SU(2) Chern-Simons 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 Chern-Simons integration cycle.
Highlights
The intersection of mathematical phenomenology and physical mathematics has historically led to profound insights into mathematical structures and the natural world
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
Inspired by the observations of [18] concerning the analytic continuation of Chern-Simons theory that we briefly summarized in section 2, we approach this task by evaluating the Jones polynomial at various roots of unity
Summary
The intersection of mathematical phenomenology and physical mathematics has historically led to profound insights into mathematical structures and the natural world. Though we do not have a complete explanation for why (1.2) predicts the volume so well, its implications are intriguing It points to the existence of a sort of quantum/semiclassical duality between SU(2) and SL(2, C) Chern-Simons theory, since some simple numerical coefficients are enough to transform a strong coupling object (the Jones polynomial at small k) into a weak coupling one (the hyperbolic volume). This is reminiscent of the shift k → k + 2 induced by the one-loop correction in the SU(2) Chern-Simons path integral [10], which transforms a semiclassical approximation into a more quantum result via a single O(1) parameter. Of our machine learning algorithms (appendix A), results relating the scaling of the Jones polynomial coefficients with the hyperbolic volume (appendix B), details concerning the various normalizations of the Jones polynomial in the mathematics and physics literature (appendix C), a data analysis of knot invariants using t-distributed stochastic neighbor embedding (appendix D), and an overview of related experiments (appendix E)
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