Related papers: Homology for Quandles with Partial Group Operation…
A quandle will be called quasi-affine, if it embeds into an affine quandle. Our main result is a characterization of quasi-affine quandles, by group-theoretic properties of their displacement group, by a universal algebraic condition coming…
It is well-known that the cohomology of symmetric quandles generates robust cocycle invariants for unoriented classical and surface links. Expanding on the recently introduced module-theoretic generalized cohomology for symmetric quandles,…
In this paper, we investigate the structure of associated groups of symmetric quandles. Among other results, we explore the relationship between the associated group of a symmetric quandle and that of its underlying quandle. We provide a…
We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most…
We propose a simple method to produce quandle cocycles from group cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between…
We introduce the notion of a hierarchical quandle, which is a generalisation of diquandles and multi-quandles. Using hierarchical quandle colourings, we construct a cocycle invariants for links coloured by quandles.
The knot group is the fundamental group of a knot or link complement. A necessary and sufficient conditions for a group to be realized as the knot group of some link was provided. This result was shown using the closed braid method.…
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…
In this paper we define the fundamental quandle of knotoids and linkoids and prove that it is invariant under the under forbidden-move and hence encodes only the information of the underclosure of the knotoid. We then introduce $n$-pointed…
In the paper we describe the class of principal quandles and show that connected quandles can be decomposed as a disjoint union of principal quandles. We also prove that simple affine quandles are finite and they can be characterized among…
In CJKLS quandle cohomology is used to produce invariants for particular embeddings of codimension two; 2-cocycles give to invariants for (classical) knots and 3-cocycles give rise to invariants for knotted surfaces. This is done by way of…
In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under two types of cube diagram operations. A knot…
Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions…
When a boudnary-parabolic representation of a link group to PSL(2,$\mathbb{C}$) is given, Inoue and Kabaya suggested a combinatorial method to obtain the developing map of the representation using the octahedral triangulation and the…
This paper introduces a homology theory for links in I-bundles over an orientable surface. The theory is unique in that the elements of the chain groups are surfaces instead of diagrams. It is then shown this theory yields the same results…
Quandles can be regarded as generalizations of symmetric spaces. In the study of symmetric spaces, the notion of flatness plays an important role. In this paper, we define the notion of flat quandles, by referring to the theory of…
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…
We define the notion of the orbit group of a quandle via its connectivity and compute the orbit groups for some basic quandles. We also show that the orbit group counts the number of orbits of certain quandles.
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.
Multi-virtual knot theory was introduced in $2024$ by the first author. In this paper, we initiate the study of algebraic invariants of multi-virtual links. After determining a generating set of (oriented) multi-virtual Reidemeister moves,…