Dmitriy Derevnin | Searcharxiv

arXiv2004

Volumes for twist link cone-manifolds

Recently, the explicit volume formulae for hyperbolic cone-manifolds, whose underlying space is the 3-sphere and the singular set is the knot $4_1$ and the links $5^2_1$ and $6^2_2$, have been obtained by the second named author and his collaborators. In this paper we explicitly find the hyperbolic volume for cone-manifolds with the link $6^2_3$ as singular set. Trigonometric identities (Tangent, Sine and Cosine Rules) between complex lengths of singular components and cone angles are obtained for an infinite family of two-bridge links containing $5^2_1$ and $6^2_3$.

math.GT↗

arXiv2002

On the volume of spherical Lambert cube

The calculation of volumes of polyhedra in the three-dimensional Euclidean, spherical and hyperbolic spaces is very old and difficult problem. In particular, an elementary formula for volume of non-euclidean simplex is still unknown. One of the simplest polyhedra is the Lambert cube Q(α,β,γ). By definition, Q(α,β,γ) is a combinatorial cube, with dihedral angles α,βand γassigned to the three mutually non-coplanar edges and right angles to the remaining. The hyperbolic volume of Lambert cube was found by Ruth Kellerhals (1989) in terms of the Lobachevsky function Λ(x). In the present paper the spherical volume of Q(α,β,γ) is defined in the terms of the function δ(α,θ) which can be considered as a spherical analog of the Lobachevsky function Δ(α,θ)=Λ(α+ θ) - Λ(α- θ)

math.MG↗