Related papers: The Topological Quandles up to Four Elements

We characterize finite $p$-groups $G$ of order up to $p^7$ for which the group of central automorphisms fixing the center element-wise is of minimum possibe order.

It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on…

The fundamental quandle is a complete invariant for unoriented tame knots \cite{JO, Ma} and non-split links \cite{FR}. The proof involves proving a relationship between the components of the fundamental quandle and the cosets of the…

In generalization of knot quandles we introduce similar algebraic structures associated with arbitrary pairs consisting of a path-connected topological space and its path-connected subspace.

We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces,…

Let $\Gamma$ be a discrete group of finite virtual cohomological dimension with certain finiteness conditions of the type satisfied by arithmetic groups. We define a representation ring for $\Gamma$, determined on its elements of finite…

We introduce a notion of topological quandle. Given a topological quandle $Q$ we associate to every classical link $L$ in $\R ^3$ an invariant $J_Q(L)$ which is a topological space (defined up to a homeomorphism). The space $J_Q(L)$ can be…

Is the universe finite or infinite, and what shape does it have? These fundamental questions, of which relatively little is known, are typically studied within the context of the standard model of cosmology where the universe is assumed to…

Consider a finite, regular cover $Y\to X$ of finite graphs, with associated deck group $G$. We relate the topology of the cover to the structure of $H_1(Y;\mathbb{C})$ as a $G$-representation. A central object in this study is the {\em…

In this paper we provide an alternative characterization of finite simply connected quandles involving only cocycles with values in abelian groups of prime size. As a corollary of such a characterization and the classification of connected…

In this paper we describe three geometric applications of quandle homology. We show that it gives obstructions to tangle embeddings, provides the lower bound for the 4-move distance between links, and can be used in determining periodicity…

For finitary regular monads T on locally finitely presentable categories we characterize the finitely presentable objects in the category of T-algebras in the style known from general algebra: they are precisely the algebras presentable by…

Without leaving finite mathematics and using finite topological spaces only, we give a definition of homeomorphisms of finite abstract simplicial complexes or finite graphs. Besides exploring the definition in various contexts, we add some…

The Dehn quandle of a closed orientable surface is the set of isotopy classes of non-separating simple closed curves with a natural quandle structure arising from Dehn twists. In this paper, we consider finiteness of some canonical…

In this article, we define quasiprimitive quandles and describe them with the help of quasiprimitive permutation groups. As a consequence, we enumerate finite non-affine simple quandles up to order $4096$.

We obtain the formula computing the number of isomorphic classes of element systems with characters over finite commutative group $G$.

We observe that many of the separation axioms of topology (including $T_0-T_4$) can be expressed concisely and uniformly in terms of category theory as lifting properties (in the sense of Quillen model categories) with respect to (usually…

We show that the concept of topological order, introduced to describe ordered quantum systems which cannot be classified by broken symmetries, also applies to classical systems. Starting from a specific example, we show how to use pure…

To better understand the fundamental quandle of a knot or link, it can be useful to look at finite quotients of the quandle. One such quotient is the $n$-quandle (or, when $n=2$, the {\em involutory} quandle). Hoste and Shanahan \cite{HS2}…

In this article we study homotopes of finite-dimensional algebras (not necessarily, associative). In the case of associative algebras we study homotopes by methods of Category theory and give description of so-called well-tempered elements…