2025/05/14 by Peter Doyle, Doyle, Peter, Matthew Ellison +3
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Geometric and Algebraic Topology #Digital Image Processing Techniques
paper · pdf · doi:10.48550/arxiv.2505.09736
Let σ be a simplicial triangulation of the 2-sphere, X the associated integral 2-cycle. A filling of X is an integral 3-chain M with ∂ M = X; a taut filling is one with minimal L1-norm. We show that any taut filling arises from an extension of σ to a simplicial complex homeomorphic to the 3-ball. The filling is clean: it has no repeated tetrahedron, and its support complex is a clean simplicial complex. This support complex is shellable and flag: every clique in its 1-skeleton occurs as a simplex. The key to the proof is the general fact that any taut filling of an n-cycle splits under disjoint union, connected sum, and more generally what we call almost disjoint union, where summands are supported on sets that overlap in at most n+1 vertices. We used AI to formalize and prove in Lean the splitting theorem and the resulting cleanness, shellability, and flagness results.