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May 25, 2026 · math.DGJ/K move · Enter open · S save

Minimal surfaces, Knots, and Neural Networks

Tancredi Schettini Gherardini, Marco Usula

University of Bonn and Max Planck Institute for Mathematics, Bonn, Germany. · Mathematical Institute, University of Oxford, Oxford, United Kingdom.

A recent conjecture by Joel Fine posits a relationship between the coefficients of the HOMFLY polynomial of a knot

K

in the 3-sphere

S^3

, and the signed count of minimal surfaces in hyperbolic 4-space

\mathrm{H}^4

meeting the sphere at infinity at

K

, with prescribed genus and self-intersection number. In this paper, we develop a novel machine learning framework based on Physics-Informed Neural Networks (PINNs) to solve the minimal surface equation in hyperbolic space. We utilise this framework to test Fine's Conjecture by constructing near-minimal surfaces bounding various families of knots in

S^3

. Furthermore, we develop an algorithmic method to find self-intersections and compute their sign. For every knot analysed, the computationally discovered minimal surfaces and their self-intersection numbers perfectly align with the predictions of Fine's Conjecture, providing empirical evidence for it.

Knots Conventional Physics-Informed Neural Networks Neural Network Polynomials

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