arXiv subjects

Selçuk Kayacan

Publications and source records attributed to Selçuk Kayacan.

10 recordsLinked to original sources

arXiv2026

Finite Topological Space Filtrations: A Topological Framework for Data Analysis

We introduce a data-analysis framework based on filtrations of finite topological spaces. Starting from a finite metric data set, we construct a sequence of coarsening topologies on the same set of points. These topologies give persistence modules and barcodes in the usual way, but they also retain information that is lost when the filtration is reduced to homology. At each level one can examine, for example, which points are topologically indistinguishable, how their minimal neighbourhoods overlap, how connected components merge, and how these features change from one level to the next. We develop the basic theory of these filtrations, establish stability results under suitable hypotheses, and give practical constructions starting directly from a distance matrix. We then study what can be learned from the resulting finite topologies. On synthetic data with known clusters of different shapes, sizes, and densities, we examine how these regions appear among the finite-topological structures and how they merge as the topology coarsens. We also study what happens when points that become uncovered early in the construction are removed and the analysis is repeated. For one-dimensional homology, we use paths in the finite-topological structure to locate cycles and to examine how their appearance is related to the geometry of the data. We finally apply these ideas to two real data sets with quite different structures. On the Paul15 single-cell data, we use the evolving finite topology to examine fine cellular states, their overlaps and relations, their assembly into larger groups, and the effect of removing points that connect these structures. On COIL20, where images of an object are sampled through a full rotation, we study how the cyclic organization of the images is reflected in the finite-topological evolution and in the associated one-dimensional homology.

math.AT↗

arXiv2026

Subrack lattices of finite solvable and metacyclic groups

A group $G$ with conjugation operation is a rack. We call such racks \emph{group racks}. In this paper we study finite group racks via their subrack lattices. Heckenberger, Shareshian, and Welker proved that the isomorphism type of the subrack lattice of a finite group determines whether the group is solvable. Our first result shows that if $G$ is a finite solvable group and $H$ is a finite group whose subrack lattice is isomorphic to that of $G$, then $H$ is solvable and the derived length of $H$ has the same derived length as $G$. Our second result is that if $G$ is a finite metacyclic group and $H$ is a group whose subrack lattice is isomorphic to that of $G$, then $H/Z(H)$ is metacyclic. As a further application of our analysis of finite metacyclic groups, we answer a question of Heckenberger, Shareshian, and Welker in the affirmative by constructing two finite groups with isomorphic subrack lattices that are not isomorphic as racks.

math.GR↗

arXiv2025

Presentations of Racks

Presentations of racks is studied and a cryptographic protocol defined on racks is proposed.

cs.CR↗

arXiv2024

Subrack Lattices of Conjugation Racks

A rack is a set with a binary operation such that left multiplications are automorphisms of the set and a quandle is a rack satisfying a certain condition. Let $S$ be a subset of a finite group $G$ which is closed under the conjugation operation $a\triangleright b := aba^{-1}$. The set $S$ with the conjugation operation $\triangleright$ is a quandle. We call those objects \emph{conjugation racks}. The prime examples are \begin{itemize} \item the group rack $(G,\triangleright)$, \item the conjugacy class rack $(C,\triangleright)$, where $C$ is a conjugacy class in $G$, and \item the $p$-power rack $(G_p,\triangleright)$, where $p$ is a prime and $G_p$ is the set of all elements in $G$ whose order is a power of $p$. \end{itemize} The set of all subracks of a finite rack form a lattice under inclusion. In this paper we study the subrack lattices of the conjugation racks. In particular, we show that the subrack lattice can be associated with a subposet of a partition lattice as well as with a subposet of an integer partition lattice in a canonical way if the rack is connected. And, if the rack is not connected, the study of the homotopy properties of the subrack lattice can be reduced into the study of the homotopy properties of the subposet of parabolic subracks. We also prove that for a certain class of $p$-power racks the order of a Sylow $p$-subgroup divides the reduced Euler characteristic of the subrack lattice of the $p$-power rack. This statement can be considered as the rack analogue of a result by Brown in the field of subgroup complexes regarding the Euler characteristic of the poset of nontrivial $p$-subgroups of a group.

math.GR↗

arXiv2022

Some remarks on the subrack lattice of finite racks

The set of all subracks $\mathcal{R}(X)$ of a finite rack $X$ form a lattice under inclusion. We prove that if a rack $X$ satisfies a certain condition then the homotopy type of the order complex of $\mathcal{R}(X)$ is a $(m-2)$-sphere, where $m$ is the number of maximal subracks of $X$. The rack $X$ satisfying the condition of this general result is necessarily decomposable. Two particular instances occur when \begin{itemize} \item $X=G$ is a group rack, and when \item $X=C$ is a conjugacy class rack of a nilpotent group. \end{itemize} We also studied the subrack lattices of indecomposable racks by focusing on the conjugacy class racks of symmetric or alternating groups and determined the homotopy types of the corresponding order complexes in some cases.

math.GR↗

arXiv2021

On a conjecture about profiles of finite connected racks

A rack is a set with a binary operation such that left multiplications are automorphisms of the set and a quandle is a rack satisfying a certain condition. For a finite connected rack the cycle type of the permutation defined by left multiplication by an element is independent from the chosen element. This cycle type is called the profile of the rack. Hayashi conjectured, in the profile of a finite connected quandle, the length of a cycle must divide the length of the largest cycle. In this paper, we prove Hayashi's Conjecture in some particular cases.

math.GR↗

arXiv2020

Recovering information about a finite group from its subrack lattice

We prove that the isomorphism type of the subrack lattice of a finite group determines the nilpotence class. We analyze the problem of estimating the orders of the group elements corresponding to the atoms of the subrack lattice. As a result, we show that the subrack lattice determines $p$-nilpotence of the group if a certain condition is met.

math.GR↗

arXiv2018

Dominating Sets in Intersection Graphs of Finite Groups

Let $G$ be a group. The intersection graph $Γ(G)$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$ where $1$ denotes the trivial subgroup of $G$. In this paper we studied the dominating sets in intersection graphs of finite groups. It turns out a subset of the vertex set is a dominating set if and only if the union of the corresponding subgroups contains the union of all minimal subgroups. We classified abelian groups by their domination number and find upper bounds for some specific classes of groups. Subgroup intersection is related with Burnside rings. We introduce the notion of intersection graph of a $G$-set (somewhat generalizing the ordinary definition of intersection graph of a group) and establish a general upper bound for the domination number of $Γ(G)$ in terms of subgroups satisfying a certain property in Burnside ring. Intersection graph of $G$ is the $1$-skeleton of the simplicial complex whose faces are the sets of proper subgroups which intersect non-trivially. We call this simplicial complex intersection complex of $G$ and show that it shares the same homotopy type with the order complex of proper non-trivial subgroups of $G$. We also prove that if domination number of $Γ(G)$ is $1$, then intersection complex of $G$ is contractible.

math.GR↗

arXiv2016

Connectivity of Intersection Graphs of Finite Groups

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$ where $1$ denotes the trivial subgroup of $G$. In this paper, we classify finite solvable groups whose intersection graphs are not $2$-connected and finite nilpotent groups whose intersection graphs are not $3$-connected. Our methods are elementary.

math.GR↗

arXiv2016

$K_{3,3}$-free Intersection Graphs of Finite Groups

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$ where $1$ denotes the trivial subgroup of $G$. In this paper we classify all finite groups whose intersection graphs are $K_{3,3}$-free.

math.GR↗