Taut fillings

TL;DR

This paper establishes a precise equivalence between minimal tetrahedral extensions of sphere triangulations and minimal integral 3-chains, revealing structural properties of optimal fillings and their shellability.

Contribution

It proves that the minimal number of tetrahedra needed to fill a sphere triangulation equals the minimal L1-norm of an associated 3-chain, and characterizes the structure of optimal fillings.

Findings

Zvol(σ) equals tetvol(σ)

Optimal fillings are shellable and flag complexes

Optimal fillings split under disjoint union and connected sum

Abstract

Sleator, Tarjan, and Thurston asked: Given a triangulation $\sigma$ of the 2-sphere, what is the minimum number of tetrahedra needed to extend $\sigma$ to a triangulation of the ball? Call this minimum $\mathrm{tetvol}(\sigma)$. Let $X$ be the integral 2-cycle associated to an orientation of $\sigma$, and let $\mathrm{Zvol}(\sigma)$ be the minimum $L_1$-norm of an integral 3-chain $M$ with $\partial M = X$. We show that $\mathrm{Zvol}(\sigma) = \mathrm{tetvol}(\sigma)$, and any optimal $M$ arises from an extension of $\sigma$ to a simplicial complex homeomorphic to the 3-ball. This 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 optimal filling of an integral $n$-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.