Upper Bounds for Volumes of Generalized Hyperbolic Polyhedra and Hyperbolic Links

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