arXiv:2605.26234v1 Announce Type: cross Abstract: 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 n
The paper leverages recent advancements in Physics-Informed Neural Networks (PINNs) to tackle complex mathematical conjectures that were previously difficult to test empirically.
This development indicates a growing capability for AI to contribute to theoretical mathematics and physics, potentially accelerating discoveries in fields beyond traditional computational applications.
The ability to use machine learning to test abstract mathematical conjectures suggests a new paradigm for scientific discovery, moving beyond purely human-driven intuition and proof.
- · Theoretical mathematicians
- · Physics-Informed Neural Network researchers
- · Research institutions
This framework could enable testing of numerous other complex conjectures in topology, geometry, and string theory.
New mathematical insights derived from AI assistance could lead to breakthroughs in fundamental physics and materials science.
The integration of AI into theoretical sciences might reshape academic research structures and funding priorities, emphasizing computational discovery.
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