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A. Vesnin

Publications and source records attributed to A. Vesnin.

9 recordsLinked to original sources
arXiv2026

Spectra of normalized volumes of right-angled hyperbolic polyhedra

We consider three-dimensional hyperbolic polyhedra of finite volume with finitely many vertices. The normalized volume of a polyhedron is the ratio of its volume to the number of vertices. Given some set of hyperbolic polyhedra, we can associate with it the set of normalized volumes of the polyhedra belonging to it. We call this set the spectrum of normalized volumes of the set under consideration. We focus on right-angled hyperbolic polyhedra. For the subset of ideal polyhedra, we find bounds for the spectrum of normalized volumes and prove that they are sharp. Moreover, we show that the spectrum splits into discrete and dense parts. For the subset of compact polyhedra, we obtain estimates for the spectrum of normalized volumes, prove that the upper bound is sharp, and also establish numerical intervals on which the spectrum is discrete and dense. The endpoints of the indicated numerical intervals are expressed in terms of the volume of the regular ideal hyperbolic tetrahedron and the volume of the regular ideal hyperbolic octahedron.

math.GT↗
arXiv2025

The minimal covolume right-angled Coxeter group in hyperbolic 3-space

We prove that among all right-angled Coxeter groups in hyperbolic 3-space, the group generated by reflections in the faces of a right-angled triangular bipyramid with three ideal and two finite vertices has the smallest covolume. The group is arithmetic and its covolume equals Catalan's constant G = 0.915965....

math.GT↗
arXiv2021

Virtual and universal braid groups, their quotients and representations

In the present paper we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations $B_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}]\right)$, $VB_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}, t_1^{\pm1},t_2^{\pm1},\ldots, t_{n-1}^{\pm1}]\right)$ which are connected with the famous Lawrence-Bigelow-Krammer representation. It turns out that these representations are faithful representations of crystallographic groups $B_n/P_n'$, $VB_n/VP_n'$, respectively. Using these representations we study certain properties of the groups $B_n/P_n'$, $VB_n/VP_n'$. Moreover, we construct new representations and decompositions of universal braid groups $UB_n$.

math.GR↗
arXiv2020

Volume estimates for right-angled hyperbolic polyhedra

By Andreev theorem acute-angled polyhedra of finite volume in a hyperbolic space $\mathbb H^{3}$ are uniquely determined by combinatorics of their 1-skeletons and dihedral angles. For a class of compact right-angled polyhedra and a class of ideal right-angled polyhedra estimates of volumes in terms of the number of vertices were obtained by Atkinson in 2009. In the present paper upper estimates for both classes are improved.

math.GT↗
arXiv2020

Representations of flat virtual braids which do not preserve the forbidden relations

In the paper, we construct a representation $θ:FVB_n\to{\rm Aut}(F_{2n})$ of the flat virtual braid group $FVB_n$ on $n$ strands by automorphisms of the free group $F_{2n}$ with $2n$ generators which does not preserve the forbidden relations in the flat virtual braid group. This representation gives a positive answer to the problem formulated by V. Bardakov in the list of unsolved problems in virtual knot theory and combinatorial knot theory by R. Fenn, D. Ilyutko, L. Kauffman and V. Manturov. Using this representation we construct a new group invariant for flat welded links. Also we find the set of normal generators of the groups $VP_n\cap H_n$ in $VB_n$, $FVP_n\cap FH_n$ in $FVB_n$, $GVP_n\cap GH_n$ in $GVB_n$, which play an important role in the study of the kernel of the representation $θ$.

math.GR↗
arXiv2020

An unknotting invariant for welded knots

We study a local twist move on welded knots that is an analog of the virtualization move on virtual knots. Since this move is an unknotting operation we define an invariant, unknotting twist number, for welded knots. We relate the unknotting twist number with warping degree and welded unknotting number, and establish a lower bound on the twist number using Alexander quandle coloring. We also study the Gordian distance between welded knots by twist move and define the corresponding Gordian complex.

math.GT↗
arXiv2019

Ideal right-angled polyhedra in Lobachevsky space

In this paper we consider a class of right-angled polyhedra in three-dimensional Lobachevsky space, all vertices of which lie on the absolute. New upper bounds on volumes in terms the number of faces of the polyhedron are obtained. Volumes of polyhedra with at most 23 faces are computed. It is shown that the minimum volumes are realized on antiprisms and twisted antiprisms. The first 248 values of volumes of ideal right-angled polyhedra are presented. Moreover, the class of polyhedra with isolated triangles is introduces and there are obtained combinatorial bounds on their existence as well as minimal examples of such polyhedra are given.

math.GT↗
arXiv2011

Cyclic generalizations of two hyperbolic icosahedral manifolds

We discuss two families of closed orientable three-dimensional manifolds which arise as cyclic generalizations of two hyperbolic icosahedral manifolds listed by Everitt. Everitt's manifolds are cyclic coverings of the lens space $L_{3,1}$ branched over some 2-component links. We present results on covering properties, fundamental groups, and hyperbolic volumes of the manifolds belonging to these families.

math.GT↗