arXiv subjects

Andrey Egorov

Publications and source records attributed to Andrey Egorov.

6 recordsLinked to original sources

arXiv2026

Minimal-Volume Equiangular Hyperbolic $4$-Polytopes

We study finite-volume convex hyperbolic $4$-polytopes whose dihedral angles are all equal to a fixed strictly acute angle $\alpha$. Put $\alpha_0=\arccos(1/3)$ and $\alpha_1=\arccos(1/4)$. We first show that the class is empty for $\alpha<\alpha_0$. For $\alpha_0\leq\alpha<\alpha_1$, the unique polytope of minimum volume is the regular hyperbolic $4$-simplex with dihedral angle $\alpha$. As $\alpha$ increases to $\alpha_1$, this simplex degenerates to a Euclidean one and no hyperbolic simplex exists at $\alpha_1$. For $\alpha_1\leq\alpha<\pi/2$, the unique minimum is the regular equiangular hyperbolic $4$-cube. The proof combines the hyperbolic Gram--Euler relation with the Davis--Okun theorem on the Charney--Davis inequality for flag triangulations of the $3$-sphere. The geometric step is a missing-face argument showing that the boundary of the dual of every nonsimplex in the class is flag.

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arXiv2026

Minimal Equiangular Hyperbolic Polyhedra in the Tetrahedral Range

We study the minimum-volume problem for finite-volume convex hyperbolic polyhedra whose dihedral angles are all equal to a fixed number $\alpha$. In the non-obtuse case one necessarily has \[ \frac{\pi}{3}\le \alpha\le \frac{\pi}{2}. \] We prove that throughout the full range in which the regular hyperbolic tetrahedron with dihedral angle $\alpha$ exists, \[ \frac{\pi}{3}\le \alpha<\arccos\frac13, \] it is the unique minimum-volume equiangular hyperbolic polyhedron with prescribed angle $\alpha$. The left endpoint is the known ideal case, while at $\alpha=\arccos(1/3)$ the regular tetrahedron degenerates to the Euclidean one. The proof combines Andreev's theorem and the Schl\"afli formula with Atkinson's decomposition into atoroidal and prismatic parts, explicit volume estimates for ordinary prisms and complete orthoschemes, and a direct equiangular version of Inoue's edge surgery.

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arXiv2026

On lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra in dimensions from 5 to 12

We investigate lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra of finite volume. We use a geometric method of orthogonal gluings to establish new bounds in low dimensions, specifically $v_\infty(P^5) \ge 3$ and $v_{fin}(P^7) \ge 4$. By combining these initial bounds with double counting arguments and recurrence relations, we obtain improved lower bounds for both types of vertices in all higher dimensions up to $n=12$, the maximal dimension where polyhedra of this class exist.

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arXiv2024

Virtual Braids and Cluster Algebras

In 2015 Hikami and Inoue constructed a representation of the braid group in terms of cluster algebra associated with the decomposition of the complement of the corresponding knot into ideal hyperbolic tetrahedra. This representation leads to the calculation of the hyperbolic volume of the complement of the knot that is the closure of the corresponding braid. In this paper, based on the Hikami-Inoue representation discussed above, we construct a representation for the virtual braid group. We show that the so-called "forbidden relations" do not hold in the image of the resulting representation. In addition, based on the developed method, we construct representations for the flat braid group and the flat virtual braid group.

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arXiv2023

Upper bounds for volumes of generalized hyperbolic polyhedra and hyperbolic links

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.

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arXiv2020

On correlation of hyperbolic volumes of fullerenes with their properties

We observe that fullerene graphs are one-skeletons of polyhedra, which can be realized with all dihedral angles equal to $π/2$ in a hyperbolic 3-dimensional space. One of the most important invariants of such a polyhedron is its volume. We are referring this volume as a hyperbolic volume of a fullerene. It is known that some topological indices of graphs of chemical compounds serve as strong descriptors and correlate with chemical properties. We demonstrate that hyperbolic volume of fullerenes correlates with few important topological indices and so, hyperbolic volume can serve as a chemical descriptor too. The correlation between hyperbolic volume of fullerene and its Wiener index suggested few conjectures on volumes of hyperbolic polyhedra. These conjectures are confirmed for the initial list of fullerenes.

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