Minimal surfaces, Knots, and Neural Networks

A recent conjecture by Joel Fine posits a relationship between the coefficients of the HOMFLY polynomial of a knot K in the 3-sphere S3, and the signed count of minimal surfaces in hyperbolic 4-space H4 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 S3. 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.