arXiv research

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

This paper studies quandles with one non-trivial column and their properties.

We introduce the notion of quasi-triviality of quandles and define homology of quasi-trivial quandles. Quandle cocycle invariants are invariant under link-homotopy if they are associated with 2-cocycles of quasi-trivial quandles. We thus obtain a lot of numerical link-homotopy invariants.

Study on link-homotopy of pretzel links using quasi-trivial quandles and biquandles.

Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all o…

In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial $Δ_{L}(t)$ is vanishing, then $L$ admits a non-trivial coloring by any non-trivial Alexander quandle $Q$, and that if $Δ_{L}(t)=1$, then $L$ admits only the trivial coloring by any Alexa…

We study a regular closure operator in the category of quandles. We show that the regular closure operator and the pullback closure operator corresponding to the reflector from the category of quandles to its full subcategory of trivial quandles coincide, we give a simple description of this closure operator, and analy…

Classifies topological quandles on simple manifolds.

The paper explores orderability in link quandles and provides various results.

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

We study the structure of symplectic quandles, quandles which are also R-modules equipped with an antisymmetric bilinear form. We show that every finite dimensional symplectic quandle over a finite field F or arbitrary field F of characteristic other than 2 is a disjoint union of a trivial quandle and a connected quand…

Characterizes quandles with abelian inner automorphisms.

We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.

In this note we prove that for any finite quandle $X$ and any 2-cocycle $φ\in Z^2_{Q\pm}(X; \mathds{Z})$, the cocycle invariant $Φ^{\pm}_φ(K)$ is trivial for all knots $K$.

A quandle coloring obstruction prevents a specific link from being ribbon concordant.

This paper is a survey of several papers in quandle homology theory and cocycle knot invariants that have been published recently. Here we describe cocycle knot invariants that are defined in a state-sum form, quandle homology, and methods of constructing non-trivial cohomology classes.

Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial $2$-cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic condit…

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

Relations will be described between the quandle cocycle invariant and the minimal number of colors used for non-trivial Fox colorings of knots and links. In particular, a lower bound for the minimal number is given in terms of the quandle cocycle invariant.

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 of odd order dihedral quandles. For the smallest example of order 6 that is an ex…

Second part of a series on higher coverings of racks and quandles.

Lower bounds of betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of proving non-triviality of cohomology groups are also given, using virtual knots. The r…

Non-classical virtual knots may have non-isomorphic upper and lower quandles. We exploit this property to define the quandle difference invariant, which can detect non-classicality by comparing the numbers of homomorphisms into a finite quandle from a virtual knot's upper and lower quandles. The invariants for small-or…

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks a…

We classify indecomposable racks of order p^2 (p a prime). There are 2p^2 - 2p - 2 isomorphism classes, among which 2p^2 - 3p - 1 correspond to quandles. In particular, we prove that an indecomposable quandle of order p^2 is affine (=Alexander). One of the results yielding this classification is the computation of the …

Ribbon knots with isomorphic quandles are stably equivalent.

The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to gr…

Study automorphism groups of quandles up to order 10.

The paper studies quandles of hyperbolic 3-space isometries.

The study characterizes torus links' coloring quivers using dihedral quandles.

The study shows involutory quandles of certain links are not left-orderable.

We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since d…

By 2-twist-spinning the knotted graph that represents the knotted handlebody $5_2$, we obtain a knotted foam in 4-dimensional space with a non-trivial quandle cocycle invariant.

Let $G$ be a group and $\varphi \in \Aut(G)$. Then the set $G$ equipped with the binary operation $a*b=\varphi(ab^{-1})b$ gives a quandle structure on $G$, denoted by $\Alex(G, \varphi)$ and called the generalised Alexander quandle. When $G$ is additive abelian and $\varphi = -\id_G$, then $\Alex(G, \varphi)$ is the we…

The set consisting of all rotations of the Euclidean plane is equipped with a quandle structure. We show that a knot is colorable by this quandle if and only if its Alexander polynomial has a root on the unit circle in $\mathbb{C}$. Further we enumerate all non-trivial colorings of a torus knot diagram by the quandle u…

We determine the third cohomology of Alexander quandles of the form F_q[T]/(T-omega), where F_q denotes the finite field of order q and omega is an element of F-q which is neither 0 nor 1. As a result, we obtain many concrete examples of non-trivial 3-cocycles.

We prove that the degenerate part of the distributive homology of a multispindle is determined by the normalized homology. In particular, when the multispindle is a quandle $Q$, the degenerate homology of $Q$ is completely determined by the quandle homology of $Q$. For this case (and generally for two term homology of …

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

We describe a way of representing finite biquandles with n elements as 2n x 2n block matrices. Any finite biquandle defines an invariant of virtual knots through counting homomorphisms. The counting invariants of non-quandle biquandles can reveal information not present in the knot quandle, such as the non-triviality o…

The paper studies invariants of surfaces in the 3-sphere using handlebody-links.

Study of quandle coloring quivers with dihedral quandles.

Paper studies embedding conditions for homogeneous quandles.

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.

Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via graphical diagrams with virtual crossings. Virtual knot theory studies non-planar Gauss…