arXiv research

Study of profinite quandles with constructions and characterizations.

The paper extends Stone duality to topological convexity spaces.

Homology and cohomology theory for topological quandles computed.

A continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories. Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show the differences in second cohomology groups for specific topological quandles. A method of computing the c…

New colored link invariants using multi-quandles.

New algebraic structures for topological pairs.

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 interpreted as a space of colourings of a diagram of the link $L$ with colours f…

We investigate the classification of topological quandles on some simple manifolds. Precisely we classify all Alexander quandle structures, up to isomorphism, on the real line and the unit circle. For the closed unit interval $[0, 1]$, we conjecture that there exists only one topological quandle structure on it, i.e. t…

We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then…

The paper defines and calculates Euler characteristics for quandles.

The paper defines circular orderability for quandles and explores their properties.

Study fundamental quandle of ribbon concordances, proving homomorphisms.

Study on endomorphism and automorphism groups of specific quandles.

We show that there are infinitely many nonisomorphic quandle structures on any topogical space $X$ of positive dimension. In particular, we disprove the conjecture, asserting that there are no nontrivial quandle structures on the closed unit interval $[0,1]$.

The study introduces Cayley--Abels--Rosendal graphs for Polish groups.

We show that a variety of monodromy phenomena arising in geometric topology and algebraic geometry are most conveniently described in terms of quandle homomorphisms from a knot quandle associated to the base to a quandle associated to a fiber. We consider the cases of the monodromy of a branched covering, braid monodro…

Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work [2] residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles are residually finite. In this paper, we extend these resul…

Classifies homeomorphism groups of countable Stone spaces up to coarse equivalence.

The paper studies finiteness of canonical quotients in Dehn quandles of surfaces.

Enhanced coloring invariant distinguishes folded molecular chain topologies.

The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group then th…

This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles $X$, from a perspective of homotopy theory: Specifically, for any prime $\ell$ which does not divide the type of $X$, the $\ell$-torsion of this invariants is equal to a sum of the colouring po…

The paper studies knot quandles and their cohomology, proving infinite dimensionality results.

Let $X$ be a normal, separated and integral scheme of finite type over $\mathbb{Z}$ and $\mathcal{M}$ a set of closed points of $X$. To a Galois cover $\tilde{X}$ of $X$ unramified over $\mathcal{M}$, we associate a quandle whose underlying set consists of points of $\tilde{X}$ lying over $\mathcal{M}$. As the limit of…

Knot theory applied to proteins, distinguishing folded linear chains.

Complex of cuts reveals full automorphism group for certain Stone spaces.

The Stone-Weierstrass theorem aids in solving inverse problems on specific manifolds.

The paper reinterprets knot group invariants using affine transformations.

To every oriented link $L$, we associate a topologically defined biquandle $\widehat{\mathcal{B}}_{L}$, which we call the topological biquandle of $L$. The construction of $\widehat{\mathcal{B}}_{L}$ is similar to the topological description of the fundamental quandle given by Matveev. We find a presentation of the top…

This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connecte…

The abstract discusses parallels between Galois theory and Stone-Weierstrass theorem in various fields.

This paper extends rack and quandle covering theory using higher categorical Galois theory.

Study of quandle coloring quivers with dihedral quandles.

The paper explores idempotents in quandle rings and their connections to quandle coverings.

Paper studies embedding conditions for homogeneous quandles.

The paper describes topological properties of arcs and crossings in knot theory.

Symmetric quandles provide new insights into link colorings.

Hierarchical quandles extend diquandles and multi-quandles for link invariants.

New polynomial invariants from quandle action quivers.

Dehn quandles of groups and surfaces unify various quandle constructions.

This paper computes the second quandle homology group of knot n-quandles.

The paper defines conditions for good involutions in generalized Alexander quandles.

New quandles classify surfaces, with infinite cohomology.

The aim of this paper is to propose a theory of derivations for quandles. Given a quandle $A$ admitting an action by a quandle $Q$, derivations from $Q$ to $A$ are introduced as twisted analogues of quandle homomorphisms. It is shown that for each quandle $Q$ there exists a unique $Q$-quandle $\mathcal{A}_Q$ (the deriv…

New link colorings using quandle rings and idempotents are stronger than existing methods.

Alexander quandles can be embedded into groups.

The study describes good involutions in quandles and Alexander quandles.

If $A$ is an abelian quandle and $Q$ is a quandle, the hom set $\mathrm{Hom}(Q,A)$ of quandle homomorphisms from $Q$ to $A$ has a natural quandle structure. We exploit this fact to enhance the quandle counting invariant, providing an example of links with the same counting invariant values but distinguished by the hom …