Year: 2026
Type: preprint
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 3-manifold it defines. We show that this is false. Starting from the figure-eight knot complement, we construct an ideal 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.
Source: arXiv (Cornell University)
Authors: Feng, Ke, Lin, Aijin
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223006745