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Timur Nasybullov

Publications and source records attributed to Timur Nasybullov.

15 recordsLinked to original sources
arXiv2025

Comparison of addition and multiplication in a skew brace

A. Smoktunowicz and L. Vendramin conjectured that if $A=(A,\oplus,\odot)$ is a finite skew brace with solvable additive group $A_{\oplus}$, then the multiplicative group $A_{\odot}$ of $A$ is also solvable. Proving or disproving this conjecture is currently an open problem. The interest to the conjecture of A. Smoktunowicz and L. Vendramin is due to the fact that, despite the fact that the addition and multiplication in a skew brace are related to each other, they can be very different. The present work focuses on comparing addition and multiplication in a skew brace. The results presented in the paper say that if $B$ is a characteristic subgroup of $A_{\oplus}$, then under certain conditions on elements $a,b\in A$ the images of $a\odot b$ and $a\oplus b$ coincide in $A_{\oplus}/B$. As a corollary we conclude that if $A$ is a finite skew brace such that the derived subgroup $A_{\oplus}^{\prime}$ is cyclic, then $A_{\odot}$ is solvable. This statement gives a positive answer to the conjecture of A. Smoktunowicz and L. Vendramin in the case when $A_{\oplus}^{\prime}$ is a cyclic group.

math.GR↗
arXiv2020

Quandles with orbit series conditions

We introduce the notion of an orbit series in a quandle. Using this notion we define four families of quandles based on finiteness conditions on their orbit series. Intuitively, the classes tOS and tOSn correspond to finitary compositions of trivial quandles while the classes OS and OSn correspond to finitary compositions of connected quandles. We study properties of these four families of quandles and explore their relationships with several previously studied families of quandles: reductive, n-reductive, locally reductive, n-locally reductive, and solvable quandles.

math.GR↗
arXiv2020

Finite skew braces with solvable additive group

A. Smoktunowicz and L. Vendramin conjectured that if $A$ is a finite skew brace with solvable additive group, then the multiplicative group of $A$ is solvable. In this short note we make a step towards positive solution of this conjecture proving that if $A$ is a minimal finite skew brace with solvable additive group and non-solvable multiplicative group, then the multiplicative group of $A$ is not simple. On the way to obtaining this result, we prove that the conjecture of A. Smoktunowicz and L. Vendramin is correct in the case when the order of $A$ is not divisible by $3$.

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

Chevalley groups of types $B_n$, $C_n$, $D_n$ over certain fields do not possess the $R_{\infty}$-property

Let $F$ be an algebraically closed field of zero characteristic. If the transcendence degree of $F$ over $\mathbb{Q}$ is finite, then all Chevalley groups over $F$ are known to possess the $R_{\infty}$-property. If the transcendence degree of $F$ over $\mathbb{Q}$ is infinite, then Chevalley groups of type $A_n$ over $F$ do not possess the $R_{\infty}$-property. In the present paper, we consider Chevalley groups of classical series $B_n$, $C_n$, $D_n$ over $F$ in the case when the transcendence degree of $F$ over $\mathbb{Q}$ is infinite, and prove that such groups do not possess the $R_{\infty}$-property.

math.GR↗
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↗
arXiv2018

Twisted conjugacy classes in unitriangular groups

Let $R$ be an integral domain of zero characteristic. In this note we study the Reidemeister spectrum of the group ${\rm UT}_n(R)$ of unitriangular matrices over $R$. We prove that if $R^+$ is finitely generated and $n>2|R^*|$, then ${\rm UT}_n(R)$ possesses the $R_{\infty}$-property, i. e. the Reidemeister spectrum of ${\rm UT}_n(R)$ contains only $\infty$, however, if $n\leq|R^*|$, then the Reidemeister spectrum of ${\rm UT}_n(R)$ has nonempty intersection with $\mathbb{N}$. If $R$ is a field, then we prove that the Reidemeister spectrum of ${\rm UT}_n(R)$ coincides with $\{1,\infty\}$, i. e. in this case ${\rm UT}_n(R)$ does not possess the $R_{\infty}$-property.

math.GR↗
arXiv2018

On the complexity of non-orientable Seifert fibre spaces

In this paper we deal with Seifert fibre spaces, which are compact 3-manifolds admitting a foliation by circles. We give a combinatorial description for these manifolds in all the possible cases: orientable, non-orientable, closed, with boundary. Moreover, we compute a potentially sharp upper bound for their complexity in terms of the invariants of the combinatorial description, extending to the non-orientable case results by Fominykh and Wiest for the orientable case with boundary and by Martelli and Petronio for the closed orientable case.

math.GT↗
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↗
arXiv2017

Reidemeister spectrum of special and general linear groups over some fields contains 1

We prove that if $\mathbb{F}$ is an algebraically closed field of zero characteristic which has infinite transcendence degree over $\mathbb{Q}$, then there exists a field automorphism $φ$ of ${\rm SL}_n(\mathbb{F})$ and ${\rm GL}_n(\mathbb{F})$ such that $R(φ)=1$. This fact implies that ${\rm SL}_n(\mathbb{F})$ and ${\rm GL}_n(\mathbb{F})$ do not possess the $R_{\infty}$-property. However, if the transcendece degree of $\mathbb{F}$ over $\mathbb{Q}$ is finite, then ${\rm SL}_n(\mathbb{F})$ and ${\rm GL}_n(\mathbb{F})$ are known to possess the $R_{\infty}$-property.

math.GR↗
arXiv2017

On groups where the twisted conjugacy class of the unit element is a subgroup

We study groups $G$ where the $φ$-conjugacy class $[e]_φ=\{g^{-1}φ(g)~|~g\in G\}$ of the unit element is a subgroup of $G$ for every automorphism $φ$ of $G$. If $G$ has $n$ generators, then we prove that the $k$-th member of the lower central series has a finite verbal width bounded in terms of $n,k$. Moreover, we prove that if such group $G$ satisfies the descending chain condition for normal subgroups, then $G$ is nilpotent. Finally, if $G$ is a finite abelian-by-cyclic group, we construct a good upper bound of the nilpotency class of $G$.

math.GR↗
arXiv2015

The Classification of fused links

We construct the complete invariant for fused links. It is proved that the set of equivalence classes of $n$-component fused links is in one-to-one correspondence with the set of elements of the abelization $UVP_n/UVP_n^{\prime}$ up to conjugation by the elements from the symmetric group $S_n<UVB_n$.

math.GT↗