Study fundamental quandle of ribbon concordances, proving homomorphisms.
problem Understanding fundamental quandle of ribbon concordances.
method Topological definition of fundamental quandle, motion picture diagrams.
result Injective and surjective quandle homomorphisms from ribbon concordance.
The paper studies symmetries in quandles and their relative versions.
problem Understanding symmetries in quandle structures and their transformations.
method Introducing relative versions of inner automorphism and transvection groups, and using them to characterize and classify surjective homomorphisms.
result Characterization of connected homomorphisms and classification of quandle structures under certain symmetry assumptions.
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.
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…
Homomorphisms on quandle cohomology groups that raise the dimensions by one are studied in relation to the cocycle state-sum invariants of knots and knotted surfaces. Skein relations are also studied.
Study Alexander matrices for link quandles and their relation to knot invariants.
problem Understanding Alexander matrices for link quandles and their applications to knot invariants.
method Investigate f-twisted Alexander matrices and their connection to quandle cocycle invariants. result Show that f-twisted Alexander invariants of knot quandles are stronger than those of knot groups. We introduce a new homology theory of quandles, called simplicial quandle homology, which is quite different from quandle homology developed by Carter et al. We construct a homomorphism from a quandle homology group to a simplicial quandle homology group. As an application, we obtain a method for computing the complex …
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 …
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…
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…
We continue the study of the quandle of homomorphisms into a medial quandle begun in Crans and Nelson. We show that it suffices to consider only medial source quandles, and therefore the structure theorem of Jedlicka et al. provides a characterization of the Hom quandle. In the particular case when the target is 2-redu…
Alexander quandles fail to distinguish certain links, thus not detecting causality.
problem Detecting causality in spacetimes using link polynomials.
method Examined Alexander quandles' ability to distinguish specific links.
result Alexander quandles cannot distinguish the connected sum of two Hopf links and Allen-Swenberg Links.
We show that isotopy classes of simple closed curves in any oriented surface admit a quandle structure with operations induced by Dehn twists, the Dehn quandle of the surface. We further show that the monodromy of a Lefschetz fibration can be conveniently encoded as a quandle homomorphism from the knot quandle of the b…
We investigate the relationship between the quandle and biquandle coloring invariant and obtain an enhancement of the quandle and biquandle coloring invariants using biquandle structures. We also continue the study of biquandle homomorphisms into a medial biquandle begun by the second author et al., finding biquandle a…
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 paper explores relationships between quandle cohomology, extensions, and automorphisms.
problem Understanding the structure of quandle extensions and automorphisms.
method Establishes a four-term exact sequence and proves relationships involving cohomology, extensions, and automorphisms of quandles.
result Derives new relationships between quandle cohomology, extensions, and automorphisms.
The paper studies quandles of hyperbolic 3-space isometries.
problem Investigating quandles of hyperbolic 3-space isometries.
method Introducing a new quandle Q(Γ,γ) and constructing a canonical map to the conjugate quandle. result The canonical map from Q(Γ,γ) to the conjugate quandle is injective and has a discrete image. Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles. Bounding chains are realized, and used to obtain equivalence moves for homologous c…
We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of…
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…
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.
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.
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.
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.
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.
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. 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.
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.
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.
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.
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.