The volume conjecture for polyhedra implies the Stoker conjecture

TL;DR

This paper demonstrates that the Volume Conjecture for polyhedra leads to a weaker form of the Stoker Conjecture, and proves that this weak form implies the full conjecture, using dihedral angles as local coordinates.

Contribution

It establishes a logical link between the Volume Conjecture and the Stoker Conjecture through a new approach involving dihedral angles.

Findings

Volume Conjecture implies a weak version of the Stoker Conjecture.

Weak Stoker Conjecture implies the full Stoker Conjecture.

Dihedral angles serve as local coordinates for certain polyhedra.

Abstract

We show that the Volume Conjecture for polyhedra implies a weak version of the Stoker Conjecture; in turn we prove that this weak version of the Stoker conjecture implies the Stoker conjecture. The main tool used is an extension of a result of Montcouquiol and Weiss, saying that dihedral angles are local coordinates for compact polyhedra with angles $\leq \pi$.