A Minimal Irreducible Triangulation of $\mathbb{S}^{3}$

Abstract

We present a very symmetric triangulation of the 3-sphere, where every edge is in at most five facets but which is not the boundary of a polytope. This shows that not every triangulation of a sphere, where angles around faces of codimension two are less than 2π in the metric pieced together by regular Euclidean simplices, is polytopal. The counterexample presented here is the smallest triangulation of $\mathbb{S}^{3}$ where every edge is contained in an empty triangle. Moreover, it shows that a triangulation of $\mathbb{S}^{3}$ that is embeddable into $\mathbb{R}^{4}$ with straight faces is not necessarily weakly vertex-decomposable.