Upper Bounds for Volumes of Generalized Hyperbolic Polyhedra and Hyperbolic Links

Abstract

Call a polyhedron in a three-dimensional hyperbolic space generalized if finite, ideal, and truncated vertices are admitted. By Belletti’s theorem of 2021 the exact upper bound for the volumes of generalized hyperbolic polyhedra with the same one-dimensional skeleton \( \Gamma \) equals the volume of an ideal right-angled hyperbolic polyhedron whose one-dimensional skeleton is the medial graph for \( \Gamma \). We give the upper bounds for the volume of an arbitrary generalized hyperbolic polyhedron such that the bounds depend linearly on the number of edges. Moreover, we show that the bounds can be improved if the polyhedron has triangular faces and trivalent vertices. As application we obtain some new upper bounds for the volume of the complement of the hyperbolic link with more than eight twists in a diagram.