Minimal surfaces, Knots, and Neural Networks
This work provides empirical evidence for a conjectured connection between knot theory and minimal surfaces, but the approach is domain-specific and the results are computational rather than theoretical.
The authors developed a PINN-based framework to solve the minimal surface equation in hyperbolic space and tested Fine's Conjecture relating knot polynomials to signed counts of minimal surfaces. For all knots analyzed, the computed surfaces and self-intersection numbers exactly matched the conjecture's predictions.
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.