A counterexample to the geometric triangulation conjecture of Costantino, Frigerio, Martelli and Petronio

Abstract

Costantino, Frigerio, Martelli and Petronio conjectured that an ideal triangulation in which every edge has valence at least six can be realized by hyperbolic partially truncated tetrahedra on the compact manifold it defines. We show that this is false. Starting from the figure-eight knot complement, we construct a triangulation with $2688$ tetrahedra and edge valences six and eighteen whose manifold is hyperbolic with totally geodesic boundary, but which admits no such realization. The obstruction is a critical point of the covolume energy of Luo and Yang at which three tetrahedra have collapsed: convexity then excludes every geometric competitor, an Euler characteristic count excludes edges of length zero, and the critical point itself is located by a finite certificate in exact integer arithmetic. The obstruction persists in every finite cover, while a complementary theorem shows that no counterexample can have only two edge lengths or only two orbits of edges.