arXiv subjects

Andrei Vesnin

Publications and source records attributed to Andrei Vesnin.

At least 19 recordsLinked to original sources

arXiv2026

The r-coverings and local moves of planar virtual knotoids

In this paper, we study planar virtual knotoids, which generalize virtual knots and knotoids both. We consider the problem of whether two given planar virtual knotoids can be transformed into each other via a sequence of local moves and what is the shortest sequence length. By introducing $r$-covering of planar virtual knotoids, we determine the homotopy relationship between different planar virtual knotoids, and obtain lower bounds of the Gordian distance between them. On the basis of these results, we demonstrate calculations of the exact Gordian distance of several given pairs of homotopic planar virtual knotoids. Furthermore, we investigate $\Delta$-moves and virtual region crossing changes for planar virtual knotoids, and derive lower bounds of the Gordian distance with respect to both moves via $r$-coverings.

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arXiv2026

Two-variable Parity Polynomial for Virtual Knotoids

In this paper, we introduce a two-variable parity polynomial invariant for virtual knotoids, defined on oriented virtual knotoid diagrams. The construction is based on the parity of classical crossings, where each crossing is classified as even or odd and treated accordingly in the definition of the invariant. We study several fundamental properties of this invariant. We demonstrate that the parity polynomial can distinguish pairs of virtual knotoids that are not distinguished by the odd writhe, prove that it is equivalent to the affine index polynomial, and establish that it is a Vassiliev invariant of order one. Finally, we give its relationship with the Petit gluing invariant.

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arXiv2026

The unknotting numbers for plus-welded knotoids

Knotoid theory is a generalization of knot theory introduced by Turaev in 2012. In recent years, various invariants of knotoids have been studied. In this paper, we mainly discuss unknotting moves and unknotting numbers of plus-welded knotoids. Firstly, we prove that a descending diagram of a plus-welded knotoid can be transformed into a trivial one through a finite sequence of $\Omega_1$, $V\Omega_1 - V\Omega_4$, $\Omega_v$, $\Phi_{\text{over}}$, and $\Phi_+$-moves. Secondly, we extend the warping degree of knots to plus-welded knotoids and discuss its properties. Finally, by utilizing the descending diagram and the warping degree, we obtain two unknotting operations for plus-welded knotoids, referred as a crossing change and a crossing virtualization. For both operations, we find upper bounds for corresponding unknotting numbers of plus-welded knotoids.

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arXiv2025

Vassiliev invariants for virtual knotoids

In this paper, we introduce the 0-smoothing invariant $\mathcal{F}$ of virtual knotoids constructed from local modification at classical crossings, which take values in a free $\mathbb Z$-module generated by non-oriented flat virtual knotoids. We prove that $\mathcal{F}$ is a Vassiliev invariant of order one. It was observed by Henrich that smoothing invariant she constructed for virtual knots provides less information than the gluing invariant. We demonstrate the same property for the 0-smoothing invariant of virtual knotoids: $\mathcal{F}$ provides less information than the gluing invariant introduced by Petit. To prove this result, we use the extension of the singular based matrix invariant originally introduced by Turaev for singular virtual strings.

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arXiv2025

Amalgamations along surfaces with boundary in a handlebody

Let M be a connected orientable 3-manifold, and F a compact connected orientable surface properly embedded in M. If F cuts M into two connected 3-manifolds X and Y, that is, M=X \cup_F Y, we say that M is an amalgamation of X and Y along F; and if F cuts M into a connected 3-manifold X, we say that M is a self-amalgamation of X along F. A characterization of an amalgamation of two handlebodies along a surface, incompressible in both, to be a handlebody was obtained by Lei, Liu, Li, and Vesnin. The case of amalgamation of two handelbodies along a compressional surface was studdied by Xu, Fang, and Lei. In the present paper, a characterization of an amalgamation and self-amalgamation of a handlebody to be a handlebody is given.

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arXiv2024

A three-variable transcendental invariant of planar knotoids via Gauss diagrams

As a generalization of the classical knots, knotoids are equivalence classes of immersions of the oriented unit interval in a surface. In recent years, a variety of invariants of spherical and planar knotoids have been constructed as extensions of invariants of classical and virtual knots. In this paper we introduce a three-variable transcendental invariant of planar knotoids which is defined over an index function of a Gauss diagram. We describe properties of this invariant and show that it is a Vassiliev invariant of order one. We also discuss the Gordian distance between planar knotoids and provide lower bounds on the Gordian distance of homotopic planar knotoids by using the transcendental invariant.

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arXiv2024

The Vol-Det Conjecture for highly twisted alternating links

The Vol-Det Conjecture, formulated by Champanerkar, Kofman and Purcell, states that there exists a specific inequality connecting the hyperbolic volume of an alternating link and its determinant. Among the classes of links for which this conjecture holds are all alternating hyperbolic knots with at most 16 crossings, 2-bridge links, and links that are closures of 3-strand braids. In the present paper, Burton's bound on the number of crossings for which the Vol-Det Conjecture holds is improved for links with more than eight twists. In addition, Stoimenow's inequalities between hyperbolic volumes and determinants are improved for alternating and alternating arborescent links with more than eight twists.

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arXiv2024

Polynomials of complete spatial graphs and Jones polynomial of related links

Let $K_n$ be a complete graph with $n$ vertices. An embedding of $K_n$ in $S^3$ is called a spatial $K_n$-graph. Knots in a spatial $K_n$-graph corresponding to simple cycles of $K_n$ are said to be constituent knots. We consider the case $n=4$. The boundary of an oriented band surface with zero Seifert form, constructed for a spatial $K_4$, is a four-component associated link. There are obtained relations between normalized Yamada and Jaeger polynomials of spatial graphs and Jones polynomials of constituent knots and the associated link.

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arXiv2023

On quasi-arithmeticity of hyperbolic gluings

We study a more general version of the gluings of hyperbolic orbifolds in the spirit of Gromov and Piatetski-Shapiro, where the gluing pieces, called the building blocks, are no longer assumed to be arithmetic or incommensurable. We prove that if such a general hyperbolic gluing along a common finite-volume totally geodesic hypersurface is quasi-arithmetic (this is a broader notion than that of arithmeticity) then each building block must be quasi-arithmetic as well and, moreover, with the same ambient group and adjoint trace field. We also show that there exist arithmetic gluings whose building blocks are incommensurable even despite the reflection with respect to the lift of the gluing locus commensurates the fundamental group of the gluing. On the other hand, we provide an example of nonarithmetic but quasi-arithmetic orbifolds such that a specific gluing of such an orbifold with itself along the boundary gives rise to an arithmetic hyperbolic orbifold. We illustrate the above results in the setting of reflection groups and hyperbolic Coxeter polyhedra and apply them to rule out the (quasi-)arithmeticity of a family of ideal hyperbolic right-angled $3$-polyhedra, namely, certain ``twisted'' ideal right-angled antiprisms, which play an important role in low-dimensional geometry and topology.

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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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arXiv2022

On volumes of hyperbolic right-angled polyhedra

In this paper we obtain new upper bounds on volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with only finite (or usual) vertices, and for finite volume polyhedra with vertices of both types.

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arXiv2021

Recurrent Generalization of F-Polynomials for Virtual Knots and Links

F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman's affine index polynomial and use smoothing in classical crossing of a virtual knot diagram. In this paper we introduce weight functions for ordered orientable virtual and flat virtual link. A flat virtual link is an equivalence class of virtual links in respect to a local symmetry changing type of classical crossing in a diagram. By considering three types of smoothings in classical crossings of a virtual link diagram and suitable weight functions, we provide a recurrent construction for new invariants. We demonstrate by providing explicit examples, that newly defined polynomial invariants are stronger than F-polynomials.

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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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arXiv2019

A characteristics for a surface sum of two handlebodies along an annulus or a once-punctured torus to be a handlebody

The main results of the paper is that we give a characteristics for an annulus sum and a once-punctured torus sum of two handlebodies to be a handlebody as follows: 1. The annulus sum $H=H_1\cup_A H_2$ of two handlebodies $H_1$ and $H_2$ is a handlebody if and only if the core curve of $A$ is a longitude for either $H_1$ or $H_2$. 2. Let $H=H_1\cup_T H_2$ be a surface sum of two handlebodies $H_1$ and $H_2$ along a once-punctured torus $T$. Suppose that $T$ is incompressible in both $H_1$ and $H_2$. Then $H$ is a handlebody if and only if the there exists a collection $\{δ, σ\}$ of simple closed curves on $T$ such that either $\{δ, σ\}$ is primitive in $H_1$ or $H_2$, or $\{δ\}$ is primitive in $H_1$ and $\{σ\}$ is primitive in $H_2$.

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arXiv2019

Polynomials of genus one prime knots of complexity at most five

Prime knots of genus one admitting diagram with at most five classical crossings were classified by Akimova and Matveev in 2014. In 2018 Kaur, Prabhakar and Vesnin introduced families of L-polynomials and F-polynomials for virtual knots which are generalizations of affine index polynomial. Here we introduce a notion of totally flat-trivial knots and demonstrate that for such knots F-polynomials and L-polynomials coincide with affine index polynomial. We prove that all Akimova - Matveev knots are totally flat-trivial and calculate their affine index polynomials.

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arXiv2019

Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $\mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high dimensional simplex in both the Gordian complexes, i.e., by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that that the constructed virtual knots have the same affine index polynomial.

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arXiv2019

$F$-polynomials of tabulated virtual knots

A sequence of $F$-polynomials $\{ F^n_K (t, \ell)\}_{n=1}^{\infty}$ of virtual knots $K$ was defined by Kaur, Prabhakar, and Vesnin in 2018. These polynomials have been expressed in terms of index value of crossing and $n$-writhe of $K$. By the construction, $F$-polynomials are generalizations of the Kauffman's Affine Index Polynomial, and are invariants of virtual knot $K$. We present values of $F$-polynomials of oriented virtual knots having at most four classical crossings in a diagram.

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arXiv2019

Structural aspects of twin and pure twin groups

The twin group $T_n$ is a Coxeter group generated by $n-1$ involutions and the pure twin group $PT_n$ is the kernel of the natural surjection of $T_n$ onto the symmetric group on $n$ letters. In this paper, we investigate structural aspects of twin and pure twin groups. We prove that the twin group $T_n$ decomposes into a free product with amalgamation for $n>4$. It is shown that the pure twin group $PT_n$ is free for $n=3,4$, and not free for $n\ge 6$. We determine a generating set for $PT_n$, and give an upper bound for its rank. We also construct a natural faithful representation of $T_4$ into $\operatorname{Aut}(F_7)$. In the end, we propose virtual and welded analogues of these groups and some directions for future work.

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