Disentangling a Deep Learned Volume Formula

TL;DR

This paper introduces a simple formula that accurately estimates the hyperbolic volume of a knot from a single evaluation of its Jones polynomial at a specific root of unity, significantly improving previous methods.

Contribution

It presents a novel phenomenological formula derived via neural network analysis that approximates knot volume with high accuracy using minimal data.

Findings

Average error of 2.86% on 1.7 million knots

Neural network trained on 10% of data achieves similar accuracy

Relevant Jones polynomial evaluations involve an analytic continuation to fractional levels

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.