Upper bounds for volumes of generalized hyperbolic polyhedra and hyperbolic links

TL;DR

This paper establishes new upper bounds for volumes of generalized hyperbolic polyhedra, improving existing bounds under certain face and vertex conditions, and applies these results to hyperbolic link complements with multiple twists.

Contribution

It provides explicit volume upper bounds for generalized hyperbolic polyhedra depending linearly on edges and refines bounds for polyhedra with triangular faces and trivalent vertices.

Findings

Derived volume bounds depending linearly on the number of edges.

Improved bounds for polyhedra with triangular faces and trivalent vertices.

New upper bounds for hyperbolic link complements with over eight twists.

Abstract

A polyhedron in a three-dimensional hyperbolic space is said to be generalized if finite, ideal and truncated vertices are admitted. In virtue of Belletti's theorem (2021) the exact upper bound for volumes of generalized hyperbolic polyhedra with the same one-dimensional skeleton $G$ is equal to the volume of an ideal right-angled hyperbolic polyhedron whose one-dimensional skeleton is the medial graph for $G$. In the present paper we give the upper bounds for the volume of an arbitrary generalized hyperbolic polyhedron, where the bonds linearly depend on the number of edges. Moreover, it is shown that the bounds can be improved if the polyhedron has triangular faces and trivalent vertices. As an application there are obtained new upper bounds for the volume of the complement to the hyperbolic link having more than eight twists in a diagram.