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 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 $e^{2\pi i/(k+2)}$ 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$\unicode{x2013}$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$\unicode{x2013}$Simons integration cycle.

arXiv commentv1: 26 + 19 pages, 15 figures; v2: 27 + 18 pages, figures updated, references added, journal version

References in corpus (20)

• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
• On Khovanov's categorification of the Jones polynomial
• Machine Learning in the String Landscape
• Machine Learning of Calabi-Yau Volumes
• Evolving neural networks with genetic algorithms to study the String Landscape
• Machine Learning CICY Threefolds
• A volume-ish theorem for the Jones polynomial of alternating knots
• Branes with Brains: Exploring String Vacua with Deep Reinforcement Learning
• Multi-Boundary Entanglement in Chern-Simons Theory and Link Invariants
• Deep Learning the Hyperbolic Volume of a Knot
• Machine Learning Line Bundle Cohomology
• Entanglement Entropy and the Colored Jones Polynomial
• Towards Novel Insights in Lattice Field Theory with Explainable Machine Learning
• Getting CICY High
• Machine Learning Calabi-Yau Four-folds
• Statistical Predictions in String Theory and Deep Generative Models
• Machine Learning and Algebraic Approaches towards Complete Matter Spectra in 4d F-theory
• Index Formulae for Line Bundle Cohomology on Complex Surfaces
• Kahler Moduli Stabilization and the Propagation of Decidability
• Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2-bridge knot

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