用神经网络验证纽结与极小曲面的数学猜想。
Minimal surfaces, Knots, and Neural Networks
- 用物理信息神经网络求解双曲空间中的极小曲面方程。
- 计算结果与费恩猜想预测完全一致，验证了纽结与曲面的对应关系。
- 适合对几何、拓扑与机器学习交叉研究感兴趣的人。
Joel Fine提出一个新猜想：三维球面 $S^3$ 中纽结 $K$ 的 HOMFLY 多项式系数，与双曲四维空间 $ m{H}^4$ 中以 $K$ 为边界、指定亏格和自交数的极小曲面的符号计数存在关联。本文提出基于物理信息神经网络（PINNs）的新方法，求解 $ m{H}^4$ 中的极小曲面方程，构造 $S^3$ 中多类纽结的近似极小曲面。同时开发算法自动检测自交点并计算其符号。所有分析的纽结中，数值计算得到的极小曲面及其自交数均与费恩猜想预测完全吻合，为该猜想提供了强有力的实证支持。
原文摘要 · Abstract (English)
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.
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