Nilpotent quandles have simple characterizations and are closely related to nilpotency.
problem Characterizing and understanding nilpotent quandles.
method Characterization of generating sets, Hopf property, construction of free nilpotent quandles, simple presentations of associated groups.
result Nilpotent quandles have the Hopf property and are equivalent to reduced peripheral systems.
Constructs the medial quandle from link's peripheral structure.
problem Building the medial quandle of links.
method From reduced Alexander module's peripheral structure.
result Medial quandle constructed successfully.
Joyce observed that the Alexander invariant and the medial quandle of a classical knot are equivalent to each other, as invariants. In the present paper, we discuss the rather complicated extension of Joyce's observation to several different medial quandles and reduced (one-variable) Alexander modules associated with c…
Study links using quandles and groups, proving key properties.
problem Understanding links through algebraic structures.
method Analyzing modules and quandles, focusing on metabelian quotients and medial quandles.
result Fundamental multivariate Alexander quandle is isomorphic to the metabelian quotient of the link group.
It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. However, it does not hold for connected quandles in general. In t…
We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from th…
In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial ΔL(t) is vanishing, then L admits a non-trivial coloring by any non-trivial Alexander quandle Q, and that if ΔL(t)=1, then L admits only the trivial coloring by any Alexa…
Improved lower bound for knot coloring using quandles.
problem Finding the minimum number of colors for knot colorings.
method Using quandles and reduced Alexander polynomials, we improved the lower bound.
result The lower bound is exactly k + 1 for L-space knots.
Study of quandle coloring quivers with dihedral quandles.
problem Link invariants and their enhancements using quandles.
method Introduced shadow quandle coloring quivers and cocycle quivers, studied equivalence with quandle coloring numbers and shadow quandle cocycle invariants.
result Equivalence of quandle coloring quivers with quandle coloring numbers and shadow quandle cocycle quivers with shadow quandle cocycle invariants for specific dihedral quandles.
The paper explores idempotents in quandle rings and their connections to quandle coverings.
problem Understanding idempotents in quandle rings and their relation to quandle coverings.
method Investigation of idempotents in quandle rings, proving properties of idempotents in free products and unions of quandles.
result Integral quandle rings of quandles of finite type that are non-trivial coverings over nice base quandles admit infinitely many non-trivial idempotents.
Paper studies embedding conditions for homogeneous quandles.
problem Embedding problem of homogeneous quandles.
method Necessary and sufficient condition for quandle homomorphisms to be embeddings.
result Generalization of embedding theorem for generalized Alexander quandles.
Symmetric quandles provide new insights into link colorings.
problem Understanding link colorings using quandles.
method Construction of symmetric quandles and isomorphism of their cohomology groups.
result Homology groups of quandles are isomorphic to those of their symmetric doubles.
Hierarchical quandles extend diquandles and multi-quandles for link invariants.
problem Constructing invariants for links colored by quandles.
method Hierarchical quandle colourings to construct cocycle invariants.
result Hierarchical quandles provide new link invariants.
New polynomial invariants from quandle action quivers.
problem Classical and virtual knot and link invariants.
method Categorification of quandle counting invariant using quandle action quivers.
result Quandle action polynomials as decategorifications.
Dehn quandles of groups and surfaces unify various quandle constructions.
problem Understanding and unifying various quandle constructions.
method Introducing Dehn quandles of groups and subsets, proving properties and embeddings.
result Dehn quandles embed naturally into their enveloping groups, and enveloping groups of certain quandles are the quandles themselves.
This paper computes the second quandle homology group of knot n-quandles.
problem Characterizing knots using quandle homology groups.
method Computation of the second quandle homology group for knot n-quandles.
result The second quandle homology group of knot n-quandles provides more information than the knot quandle's homology group.
The paper studies finiteness of canonical quotients in Dehn quandles of surfaces.
problem Finiteness of canonical quotients in Dehn quandles of surfaces.
method Analyzing the Dehn quandle structure and using quandle quotients.
result For surfaces of positive genus, the n-quandle of their Dehn quandles is finite for specific values of n. The paper defines conditions for good involutions in generalized Alexander quandles.
problem Determining conditions for good involutions in generalized Alexander quandles.
method Analyzing the structure of generalized Alexander quandles and their involutions.
result Classification of all good involutions in connected generalized Alexander quandles.
New quandles classify surfaces, with infinite cohomology.
problem Classifying closed orientable surfaces using quandles.
method Introduced new quandle families, analyzed their cohomology, and computed their idempotents.
result Second bounded quandle cohomology is infinite-dimensional.
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 AQ (the deriv…
Alexander quandles can be embedded into groups.
problem Embedding Alexander quandles into groups.
method For any twisted conjugate quandle, find a group such that the quandle is embedded into the conjugation quandle of the group.
result Alexander quandles can be embedded into groups.
New link colorings using quandle rings and idempotents are stronger than existing methods.
problem Improving the quandle coloring invariant of links.
method Using quandle rings and their idempotents to enhance the quandle coloring invariant.
result The new invariants are stronger than the $\Hom$ quandle invariant for certain coloring quandles.
The study describes good involutions in quandles and Alexander quandles.
problem Characterizing and enumerating good involutions in quandles and Alexander quandles.
method Completely describing good involutions of free and subquandles of twisted conjugation quandles of groups, including Alexander quandles.
result Explicit mappings for good involutions of linear quandles up to order 23.
If A is an abelian quandle and Q is a quandle, the hom set 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 …
The paper explores zero-divisors and idempotents in quandle rings, proving their absence in certain cases.
problem Understanding zero-divisors and idempotents in quandle rings.
method Development of quandle rings theory, definition of orderability, computation of idempotents, and analysis of automorphism groups.
result Quandle rings of left or right orderable quandles with semi-latin structure have no zero-divisors.
Study on quandle coloring quivers for (p, 2)-torus knots and links.
problem Understanding quandle colorings of (p, 2)-torus knots and links.
method Introduced quandle coloring quivers and studied them for dihedral quandles.
result Characterized quandle coloring quivers for (p, 2)-torus knots and links.
New symmetric quandles constructed from group elements and subgroups.
problem Understanding the structure of symmetric quandles.
method Constructing symmetric quandles from specific group elements and subgroups.
result Every symmetric quandle is isomorphic to the disjoint union of constructed quandles.
Classifies spatial graphs with finite N-quandles.
problem Determining isomorphism of spatial graphs' N-quandles.
method Generalized N-quandles to spatial graphs, proving basic results and conjecturing a classification.
result Verifies conjecture in several cases, presents a possible counterexample.
In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual…
Study explores associated groups of symmetric quandles and their properties.
problem Understanding the structure and relationships of symmetric quandles and their associated groups.
method Group-theoretic analysis and characterization of associated groups.
result Characterization and properties of associated groups of symmetric quandles.
Classifies links with finite N-quandles for some N.
problem Describing finite quotients of quandles, especially n-quandles. method Introducing and studying N-quandles, proving one direction of a conjectured classification. result Proves one direction of a conjectured classification of links with finite N-quandles. This paper studies quandles with one non-trivial column and their properties.
problem Understanding quandles with exactly one non-trivially permuted column.
method Investigates automorphism groups, polynomials, cohomology, and hom quandles of quandles with one non-trivial column.
result The properties of these quandles relate to linking number.
The paper defines and analyzes quandles of knotoids and linkoids.
problem Tackling the invariant properties of knotoids and linkoids.
method Defining fundamental quandles and pointed quandles, proving invariance, and introducing new invariants.
result Fundamental pointed quandles enhance the fundamental quandle and distinguish 1-linkoids.
Study of twisted Alexander matrices for certain quandles and their invariants.
problem Investigate f-twisted Alexander matrices for quandles associated with Alexander pairs. method Define and analyze f-twisted Alexander matrices of certain quandles, relate to Carter-Saito-Satoh's invariant, and discuss connections to quandle homology groups. result 0-th elementary ideal of f-twisted Alexander matrix can be described using Carter-Saito-Satoh's invariant. 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…
Paper provides criteria to detect non-admissible quandles via coloring.
problem Determining non-admissibility of quandles.
method Using colorings of (1, 1)-tangles to detect non-admissibility.
result Constructed numerous non-admissible quandles.
The paper characterizes simply connected quandles using cocycles with prime values.
problem Characterizing simply connected quandles.
method Using cocycles with values in abelian groups of prime size.
result Classification of simply connected quandles of sizes p2 and p3 for p>3. Study of profinite quandles with constructions and characterizations.
problem Characterizing and constructing profinite quandles.
method Several constructions and characterizations of profinite quandles from profinite groups and other quandles.
result Characterization of algebraically connected profinite quandles in terms of $\widehat{\Inn(Q)}$.
New knot quandle structure for twist-spun trefoils discovered.
problem Understanding symmetries of knot-spun structures.
method Defined Schläfli quandles and studied their relationship with knot quandles.
result The knot quandle of the twist-spun trefoil is a central extension of a Schläfli quandle.
The paper defines and calculates Euler characteristics for quandles.
problem Defining and calculating Euler characteristics for quandles.
method Definition and calculation of Euler characteristics for quandles.
result The quandle Euler characteristic of a compact connected Riemannian symmetric space coincides with the topological Euler characteristic.
Homology and cohomology theory for topological quandles computed.
problem Computing invariants for knot diagrams using quandle cocycles.
method Introducing homology and cohomology theory for topological quandles, studying their relation to quandle groups, and using topological quandle cocycles to compute state sum invariants.
result State sum invariants computed using topological quandle cocycles.
The study examines Hopfian properties of conjugation quandles and their underlying groups.
problem Understanding the relationship between Hopfian properties of conjugation quandles and their underlying groups.
method Examined Hopfian and residual finiteness properties of conjugation quandles of specific groups.
result Conjugation quandles of Baumslag-Solitar groups are infinitely generated and not necessarily Hopfian.
Algorithms are described and Maple implementations are provided for finding all quandles of order n, as well as computing all homomorphisms between two finite quandles or from a finitely presented quandle (e.g., a knot quandle) to a finite quandle, computing the automorphism group of a finite quandle, etc. Several of…
New method uses quandle rings to distinguish knots and their mirrors.
problem Distinguishing knots and their mirror images.
method Combining idempotents in quandle rings with state sum invariants.
result Improved knot and mirror image distinction using fewer quandles.
Computes quandle associated groups using group homology.
problem Computing associated groups of quandles.
method Using group homology theory to describe and compute associated groups.
result Computed second quandle homology groups of specific quandle families.
In the present paper, we introduce the new construction of quandles. For a group G and its subset A we construct a quandle Q(G,A) which is called the (G,A)-quandle and study properties of this quandle. In particular, we prove that if Q is a quandle such that the natural map Q→GQ from Q to its envelopi…
Enhances knot and link invariants using quandle modules.
problem Distinguishing knots and links using polynomial invariants.
method Integrates quandle modules into the quandle coloring quiver.
result The enhanced invariant distinguishes knots and links.
Researchers describe Cayley graphs for finite involutory quandles of certain two-bridge links.
problem Understanding finite quotients of quandles for knot and link analysis.
method Explicit description of Cayley graphs for finite involutory quandles.
result Explicit descriptions of Cayley graphs for finite involutory quandles of two-bridge links with an axis.