Abstract: Volume is a useful invariant of a hyperbolic 3-manifold which can be estimated from its polyhedral decomposition. By Belletti theorem [1] the exact upper bound for the volumes of generalized hyperbolic polyhedra with the same one-dimensional skeleton G equals the volume of an ideal right-angled hyperbolic polyhedron whose one-dimensional skeleton is the medial graph for G. We will discuss the volume bounds obtained in [2] for ideal right-angled hyperbolic polyhedra and in [3] for generalized hyperbolic polyhedra. The bounds depend linearly of the number of edges of a polyhedron. As an application we get the new upper bound for volumes of hyperbolic complements of links with more than eight twists in diagrams. References: [1] G. Belletti, The maximum volume of hyperbolic polyhedra. Trans. Amer. Math. Soc. 2021, 374, 1125-1153. [2] S. Alexandrov, N. Bogachev, A. Egorov, A. Vesnin, On volumes of hyperbolic right-angled polyhedra. Sbornik: Mathematics, 2023, 214(2), 148-165. [3] A. Vesnin, A. Egorov, Upper bounds for volumes of generalized hyperbolic polyhedra and hyperbolic links. Siberian Mathematical Journal, 2024, 65(3), 469-488.

Cite: Vesnin A.
On volumes of hyperbolic polyhedra and hyperbolic knot complements
15th Serbian Mathematical Congress 19-22 Jun 2024