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Valeriy Bardakov

Publications and source records attributed to Valeriy Bardakov.

13 recordsLinked to original sources
arXiv2026

Idempotents and Powers of Ideals in Quandle Rings

This article addresses two central problems in the theory of quandle rings. First, motivated by Conjecture 3.10 in Internat. J. Math. 34 (2023), no. 3, Paper No. 2350011: for a semi-latin quandle $X$, every nonzero idempotent in the integral quandle ring $\mathbb{Z}[X]$ necessarily corresponds to an element of $X$, we investigate idempotents in quandle rings of semi-latin quandles. Precisely, we prove that if the ground ring is an integral domain with unity, then the quandle ring of Core($\mathbb{Z}$) admits only trivial idempotents. Second, powers of augmentation ideals in quandle rings have only been computed in a few cases previously. We extend the computations to include dihedral quandles and commutative quandles. Finally, we examine idempotents in quandle rings of $2$-almost latin quandles and apply these results to compute the automorphism groups of their integral quandle rings.

math.RA↗
arXiv2025

Yang-Baxter Equation and Related Algebraic Structures

In the 1990s, Drinfel'd proposed the study of set-theoretical solutions to the quantum Yang-Baxter equation, initiating a line of research that has since garnered substantial attention and led to notable developments in algebra, low-dimensional topology, and related areas. This monograph offers a concise introduction to the algebraic theory of such solutions, focusing on key structures including skew braces, quandles, racks, and Rota-Baxter groups, which have emerged as central objects in this framework. We investigate the algebraic, combinatorial, and homological properties of these structures, with an emphasis on their interrelations and applications to knot theory. The monograph is intended as a reference for researchers interested in the deep interplay between these algebraic structures and the quantum Yang-Baxter equation.

math.QA↗
arXiv2023

Links and dynamics

Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the equivalence class of the Hopf knot, which is the orbit space of the unstable saddle separatrix in the manifold $\mathbb{S}^2\times \mathbb{S}^1$, is a complete invariant of the topological conjugacy of the system. In this paper we distinguish a class of three-dimensional Morse-Smale diffeomorphisms for which the complete invariant of topological conjugacy is the equivalence class of a link in $\mathbb{S}^2\times \mathbb{S}^1$. We proved that if $M$ is a link complement in $\mathbb{S}^3$ (in particular, is $\mathbb{S}^3$), or a handlebody $H_g$ of genus $g \geq 0$, or closed, connected, orientable 3-manifold, then the set of equivalence classes of tame links in $M$ is countable. As corollary we get that in $\mathbb{S}^2\times \mathbb{S}^1$ there exists a countable number of equivalence classes of tame links. It is proved that any essential link can be realized by a diffeomorphism of the class under consideration.

math.DS↗
arXiv2021

Quandle cohomology, extensions and automorphisms

The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles.

math.GT↗
arXiv2021

A Wells type exact sequence for non-degenerate unitary solutions of the Yang--Baxter equation

Cycle sets are known to give non-degenerate unitary solutions of the Yang--Baxter equation and linear cycle sets are enriched versions of these algebraic systems. The paper explores the recently developed cohomology and extension theory for linear cycle sets. We derive a four term exact sequence relating 1-cocycles, second cohomology and certain groups of automorphisms arising from central extensions of linear cycle sets. This is an analogue of a similar exact sequence for group extensions known due to Wells. We also compare the exact sequence for linear cycle sets with that for their underlying abelian groups via the forgetful functor and discuss generalities on dynamical 2-cocycles.

math.GR↗
arXiv2020

Multi-switches and virtual knot invariants

In the paper we introduce a general approach how for a given virtual biquandle multi-switch $(S,V)$ on an algebraic system $X$ (from some category) and a given virtual link $L$ construct an algebraic system $X_{S,V}(L)$ (from the same category) which is an invariant of $L$. As a corollary we introduce a new quandle invariant for virtual links which generalizes previously known quandle invariants for virtual links.

math.AT↗
arXiv2019

General constructions of biquandles and their symmetries

Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set-theoretic solutions of the well-known Yang-Baxter equation. The first half of this paper proposes some natural constructions of biquandles from groups and from their simpler counterparts, namely, quandles. We completely determine all words in the free group on two generators that give rise to (bi)quandle structures on all groups. We give some novel constructions of biquandles on unions and products of quandles, including what we refer as the holomorph biquandle of a quandle. These constructions give a wealth of solutions of the Yang-Baxter equation. We also show that for nice quandle coverings a biquandle structure on the base can be lifted to a biquandle structure on the covering. In the second half of the paper, we determine automorphism groups of these biquandles in terms of associated quandles showing elegant relationships between the symmetries of the underlying structures.

math.GR↗
arXiv2019

Multi-switches and representations of braid groups

In the paper, we introduce the notion of a (virtual) multi-switch which generalizes the notion of a (virtual) switch. Using (virtual) multi-switches we introduce a general approach on how to construct representations of (virtual) braid groups by automorphisms of algebraic systems. As a corollary, we introduce new representations of virtual braid groups which generalize several previously known representations.

math.GR↗
arXiv2019

On embeddings of quandles into groups

In the present paper, we introduce the new construction of quandles. For a group $G$ and its subset $A$ we construct a quandle $Q(G,A)$ which is called the $(G,A)$-quandle and study properties of this quandle. In particular, we prove that if $Q$ is a quandle such that the natural map $Q\to G_Q$ from $Q$ to its enveloping group $G_Q$ is injective, then $Q$ is the $(G,A)$-quandle for an appropriate group $G$ and its subset $A$. Also we introduce the free product of quandles and study this construction for $(G,A)$-quandles. In addition, we classify all finite quandles with enveloping group $\mathbb{Z}^2$.

math.GR↗
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.

math.GR↗
arXiv2017

Automorphism groups of quandles and related groups

In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle $Q={\rm Conj}(G)$ of a group $G$ we determine several subgroups of ${\rm Aut}(Q)$ and find necessary and sufficient conditions when these subgroups coincide with the whole group ${\rm Aut}(Q)$. In particular, we prove that ${\rm Aut}({\rm Conj}(G))={\rm Z}(G)\rtimes {\rm Aut}(G)$ if and only if either ${\rm Z}(G)=1$ or $G$ is one of the groups $\mathbb{Z}_2$, $\mathbb{Z}_2^2$ or $\mathbb{Z}_3$. For a big list of Takasaki quandles $T(G)$ of an abelian group $G$ with $2$-torsion we prove that the group of inner automorphisms ${\rm Inn}(T(G))$ is a Coxeter group. We study automorphisms of certain extensions of quandles and determine some interesting subgroups of the automorphism groups of these quandles. Also we classify finite quandles $Q$ with $3\leq k$-transitive action of ${\rm Aut}(Q)$.

math.GR↗
arXiv2012

Class preserving automorphisms of unitriangular groups

Let $\textrm{UT}_n (K)$ be a unitriangular group over a field $K$ and $Γ_{n,k} := \textrm{UT}_n (K)/ γ_k(\textrm{UT}_n (K))$, where $γ_k (\mathrm{UT}_n(K))$ denotes the $k$-th term of the lower central series of $\mathrm{UT}_n (K)$, $2 \le k \le n$. We prove that the group of all class preserving automorphisms of $Γ_{n,k}$ is equal to $\Inn(Γ_{n,k})$ if and only if $K$ is a prime field. Let $G_n^{(m)} := \mathrm{UT}_n (\mathbb{F}_{p^m}) / γ_3 (\mathrm{UT}_n(\mathbb{F}_{p^m}))$. We calculate the group of all class preserving automorphisms and class preserving outer automorphisms of $G_n^{(m)}$.

math.GR↗