Taut fillings

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.