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V. G. Bardakov

Publications and source records attributed to V. G. Bardakov.

16 recordsLinked to original sources
arXiv2025

Invariants of Handlebody-Links and Spatial Graphs

A $G-$family of quandles is an algebraic construction which was proposed by A. Ishii, M. Iwakiri, Y. Jang, K. Oshiro in 2013. The axioms of these algebraic systems were motivated by handlebody-knot theory. In the present work we investigate possible constructions which generalise $G-$family of quandles and other similar constructions (for example, $Q-$ and $(G,*,f)-$families of quandles). We provide the necessary conditions under which the resulting object (called an $(X,G,{*_g},f,\otimes,\oplus)-$system) gives a colouring invariant of knotted handlebodies. We also discuss several other modifications of the proposed construction, providing invariants of spatial graphs with an arbitrary (finite) set of values of vertex valency. Besides, we consider several examples which in particular showcase the differences between spatial trivalent graph and handlebody-link theories.

math.GT↗
arXiv2025

Orders of Products of Slanted Class Transpositions

In the present work, we continue the research initiated in the preprint: V. G. Bardakov, A. L. Iskra, Orders of products of horizontal class transpositions, arXiv:2409.13341, and related to S. Kohl's question on the orders of products of pairs of class transpositions. In the preprint, an answer was given to S. Kohl's question for horizontal class transpositions. In the present work, the products of pairs of slanted class transpositions are considered, and under certain conditions, their orders are determined, with it being established that the number of different orders is finite.

math.GR↗
arXiv2024

Orders of products of horizontal class transpositions

The class transposition group $CT(\mathbb{Z})$ was introduced by S. Kohl in 2010. It is a countable subgroup of the permutation group $Sym(\mathbb{Z})$ of the set of integers $\mathbb{Z}$. We study products of two class transpositions $CT(\mathbb{Z})$ and give a partial answer to the question 18.48 posed by S. Kohl in the Kourovka notebook. We prove that in the group $CT_{\infty}$, which is a subgroup of $CT(\mathbb{Z})$ and generated by horizontal class transpositions, the order of the product of a pair of horizontal class transpositions belongs to the set $\{1,2,3,4,6,12\}$, and any number from this set is the order of the product of a pair of horizontal class transpositions.

math.GR↗
arXiv2024

Some properties of relative Rota--Baxter operators on groups

We find connection between relative Rota--Baxter operators and usual Rota--Baxter operators. We prove that any relative Rota--Baxter operator on a group $H$ with respect to $(G, Ψ)$ defines a Rota--Baxter operator on the semi-direct product $H\rtimes_Ψ G$. On the other side, we give condition under which a Rota--Baxter operator on the semi-direct product $H\rtimes_Ψ G$ defines a relative Rota--Baxter operator on $H$ with respect to $(G, Ψ)$. We introduce homomorphic post-groups and find their connection with $λ$-homomorphic skew left braces. Further, we construct post-group on arbitrary group and a family post-groups which depends on integer parameter on any two-step nilpotent group. We find all verbal solutions of the quantum Yang-Baxter equation on two-step nilpotent group.

math.GR↗
arXiv2024

Rota--Baxter and averaging operators on racks and rack algebras

In the present article we define and investigate relative Rota--Baxter operators and relative averaging operators on racks and rack algebras. Also, if B is a Rota--Baxter or averaging operator on a rack X, then we can extend B by linearity to the rack algebra k[X]. On the other side, we have definitions of Rota--Baxter and averaging operators on arbitrary algebra. We find connections between these operators. In particular, we prove that if B : X --> X is an averaging operator on a rack, then its linear extension on a rack algebra k[X] gives an averaging operator.

math.RA↗
arXiv2021

On residually nilpotence of group extensions

We study the following question: under what conditions extension of one residually nilpotent group by another residually nilpotent group is residually nilpotent? We prove some sufficient conditions under which this extension is residually nilpotent. Also, we study this question for semi-direct products and, in particular, for extensions of free group by infinite cyclic group: $F_n \rtimes_φ \mathbb{Z}$. We find conditions under which this group is residually nilpotent, find conditions under which this group has long lower central series. In particular, we prove that for $n=2$ the length of the lower central series of $F_n \rtimes_φ \mathbb{Z}$ is equal to 2, $ω$ or $ω^2$.

math.GR↗
arXiv2021

On homotopy braids

Homotopy braid group is the subject of the paper. First, linearity of homotopy braid group over the integers is proved. Then we prove that the group homotopy braid group on three strands is torsion free.

math.GR↗
arXiv2020

On the lower central series of Baumslag-Solitar groups

We find the lower central series for residually nilpotent Baumslag-Solitar groups, and find the intersection of all terms of the lower central series. Also, we find non-abelian Bauslag-Solitar groups for which the lower central series has length 2. For some Baumslag-Solitar groups a connection is found between the intersection of all subgroups of finite index and the intersection of all terms of the lower central series.

math.GR↗
arXiv2018

Groups of virtual trefoil and Kishino knots

In the paper of Yu. A. Mikhalchishina for an arbitrary virtual link $L$ three groups $G_{1,r}(L)$, $r>0$, $G_{2}(L)$ and $G_{3}(L)$ were defined. In the present paper these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to each other is proved. Also we prove that $G_3$ distinguishes the Kishino knot from the trivial knot. The fact that these groups have the lower central series which does not stabilize on the second term is noted. Hence we have a possibility to study these groups using quotients by terms of the lower central series and to construct representations of these groups in rings of formal power series. It allows to construct an invariants for virtual knots.

math.GT↗
arXiv2016

Representations of virtual braids by automorphisms and virtual knot groups

In the present paper the representation of the virtual braid group $VB_n$ into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones. The fact that these already known representations are not faithful for $n \geq 4$ is verified. Using representations of $VB_n$, the virtual link group is defined. Also representations of welded braid group $WB_n$ are constructed and the welded link group is defined.

math.AT↗
arXiv2013

Twisted conjugacy classes of the unit element

We study twisted conjugacy classes of the unit element in different groups. Fel'shtyn and Troitsky showed that the twisted conjugacy class of the unit element of an abelian group is a subgroup for every automorphism. The structure is investigated of a group whose twisted conjugacy class of the unit element is a subgroup for every automorphism (inner automorphism).

math.GR↗
arXiv2010

Brunnian Braids on Surfaces

We determine a set of generators for the Brunnian braids on a general surface $M$ for $M\not=S^2$ or $\RP^2$. For the case $M=S^2$ or $\RP^2$, a set of generators for the Brunnian braids on $M$ is given by our generating set together with the homotopy groups of a 2-sphere.

math.GT↗
arXiv2009

On the pure virtual braid group $PV_3$

In this article, we investigate various properties of the pure virtual braid group PV_3. From its canonical presentation, we obtain a free product decomposition of PV_3. As a consequence, we show that PV_3 is residually torsion free nilpotent, which implies that the set of finite type invariants in the sense of Goussarov-Polyak-Viro is complete for virtual pure braids with three strands. Moreover we prove that the presentation of PV_3 is aspherical. Finally we determine the cohomology ring and the associated graded Lie algebra of PV_3.

math.GT↗
arXiv2006

On Linear Representations of Some Extensions

Yu. I. Merzljakov developed a method of splittable coordinates which helps to verify the linearity of some groups, he established some fundamental results using this method. In this paper we use the method of splittable coordinates and find some sufficient condition under which the semi--direct product of two linear groups is linear. As consequence we get linearity of some HNN-extensions of a free group, linearity of the holomorph of the braid group B_n, n >1, and linearity of some Artin groups. In all cases we construct faithful linear representations in the explicit form.

math.GR↗
arXiv2003

The structure of the group of conjugating automorphisms and the linear representation of the braid groups of some manifolds

In this paper we describe the structure of a group of conjugating automorphisms $C_n$ of free group and prove that this structure is similar to the structure of a braid group $B_n$ with $n>1$ strings. We find the linear representation of group $C_n$. Also we prove that the braid group $B_n(S^2)$ of 2--sphere, mapping class group M(0,n) of the $n$--punctured 2--sphere and the braid group $B_3(P^2)$ of the projective plane are linear. Using result of J. Dyer, E. Formanek, E. Grossman and the faithful linear representation of Lawrence--Krammer of $B_4$ we construct faithful linear representation of the automorphism group $Aut(F_2)$.

math.GR↗