A Geometric Finiteness Theory for Essential Surfaces in Knot Exteriors

What the math gives to ML

The paper provides a constructive multiscale representation principle: objects with bounded area, thickness, and ambient complexity can be encoded by finitely many lattice cells, sampled points, and quantized tangent data, with sufficiently fine codes determining the exact ambient isotopy type. This suggests a topology-preserving tokenization layer for neural networks operating on meshes, point clouds, or 3D implicit surfaces, where code resolution is selected relative to estimated reach or thickness rather than an arbitrary voxel size. The strongest practical use is to add a discrete structural branch and a consistency loss that makes representations invariant to small geometric perturbations while preserving topological distinctions. The paper also supplies scale-normalized geometric quantities that can regularize shape encoders across resolutions.