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.
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.
Paper describes how to extend multiple conjugation quandles using maps.
problem Understanding affine extensions of multiple conjugation quandles.
method Introduces augmented MCQ Alexander pairs for affine extensions.
result Affine extensions of multiple conjugation quandles can be described by quadruples of maps.
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.
Classifies good involutions in conjugation subquandles and racks.
problem Classifying quandles with good involutions for applications in surface-knot theory.
method Study of subquandles of conjugation quandles, including core quandles; analysis of good involutions of faithful racks.
result Sharp bounds on the number of good involutions of racks in these families.
A quandle is an algebra whose axioms are motivated from knot theory. A linear extension of a quandle can be described by using a pair of maps called an Alexander pair. In this paper, we show that a linear extension of a multiple conjugation quandle can be described by using a pair of maps called an MCQ Alexander pair, …
Power quandles improve group invariants and allow group presentations.
problem Understanding and classifying groups using algebraic structures.
method Introducing power quandles, which retain conjugation and power maps, and showing their effectiveness in group theory.
result Power quandles determine central quotients and centers of groups, and can approximate any group.
Embed spherical quandles into Lie groups smoothly.
problem Embedding spherical quandles into Lie groups.
method Construct smooth embeddings into conjugation quandles of Lie groups.
result Embeddings into orthogonal, Spin, or Pin groups in dimensions 1 and 3 compared with Bergman and Akita's.
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.
Quandle coloring detects causality in spacetime links.
problem Detecting causality in spacetime links using quandle colorings.
method Conjugation quandle of dihedral group D5 combined with Alexander-Conway polynomial.
result The conjugation quandle over D5 distinguishes causally related and unrelated events.
We define a functor Q from the category of multiple conjugation biquandles to that of multiple conjugation quandles. We show that for any multiple conjugation biquandle X, there is a one-to-one correspondence between the set of X-colorings and that of Q(X)-colorings diagrammatically for any …
The paper characterizes L-space 3-manifolds using quandle orderability.
problem Characterizing L-space 3-manifolds using quandle orderability.
method Using quandles and their extensions, the paper investigates orderability and circular orderability conditions.
result The n-quandle Qn(L) of a link quandle is not right circularly orderable. 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.
Two quandles from Coxeter groups studied, showing similarities in automorphism groups.
problem Understanding quandles from Coxeter groups and their automorphism groups.
method Defined two families of quandles from Coxeter quandles and Pin groups, analyzed their automorphism groups.
result Similarity in inner automorphism groups of the two quandles.
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.
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…
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. A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to group homology, and have been applied to knots and knotted surfaces. In this paper, a homology theory is defined that unifies group an…
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…
The paper extends a method to compute A-polynomials of 2-bridge knots.
problem Computing A-polynomials of 2-bridge knots.
method Generalized symplectic quandle structure and conjugation quandle equations.
result Effective computation of A-polynomials, including previously unknown ones.
The study enumerates virtual quandles up to isomorphism.
problem Classifying virtual quandles up to isomorphism.
method Computer search and classification based on conjugacy class structures of rack automorphism groups.
result Classifications of virtual racks and quandles up to order 8.
We introduce a multiple conjugation biquandle, and show that it is the universal algebra to define a semi-arc coloring invariant for handlebody-links. A multiple conjugation biquandle is a generalization of a multiple conjugation quandle. We extend the notion of n-parallel biquandle operations for any integer n, an…
The Cayley graph of quandles reveals structural properties and is studied for various classes.
problem Investigating structural properties of Cayley graphs of quandles.
method Analyzing Cayley graphs for different quandle classes and proving properties.
result Connected components of Cayley graphs of Alexander quandles correspond to cosets of specific subgroups.
We give a formula of the connected component decomposition of the Alexander quandle: Z[t±1]/(f1(t),…,fk(t))=⨆i=0a−1Orb(i), where a=gcd(f1(1),…,fk(1)). We show that the connected component Orb(i) is isomorphic to Z[t±1]/J with an expli…
Study metrics on quandles, a knot theory algebraic system.
problem Investigate metrics on quandles, a knot theory algebraic system.
method Investigate graph structures and metric spaces induced by the actions of the inner and displacement groups on quandles.
result Show that the metric space associated with the displacement group for generalized Alexander quandles is quasi-isometric to the displacement group with a word metric.
In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle Q=Conj(G) of a group G we determine several subgroups of Aut(Q) and find necessary and sufficient conditions when these subgroups coincide with the whole group Aut(Q). In particular, we p…
We categorify the theory of Lie algebras beginning with a new notion of categorified vector space, or `2-vector space', which we define as an internal category in Vect, the category of vector spaces. We then define a `semistrict Lie 2-algebra' to be a 2-vector space equipped with a skew-symmetric bilinear functor satis…
The paper introduces a new coloring invariant for spatial surfaces using a multiple group rack.
problem Distinguishing spatial surfaces embedded in the 3-sphere.
method Defined a coloring invariant using a multiple group rack.
result Introduced a new invariant to distinguish spatial surfaces.
The paper studies parabolic representations of 2-bridge links using symplectic quandles.
problem Parabolic representations of 2-bridge links.
method Convert conjugation quandle equations to symplectic quandle equations, using a polynomial PK(u) to find arc coloring vectors. result Explicit formulas for parabolic representations of 2-bridge links are derived, including complex volume and cusp shape.
This paper extends rack and quandle covering theory using higher categorical Galois theory.
problem Developing a higher covering theory of racks and quandles.
method Applying techniques from higher categorical Galois theory to extend and clarify the foundations of rack and quandle coverings.
result Identification of meaningful higher-dimensional centrality conditions defining higher coverings of racks and quandles.
For the potential function of a link diagram induced by the optimistic limit of the colored Jones polynomial, we show the existence of a solution of the hyperbolicity equations by directly constructing it. This construction is based on the shadow-coloring of the conjugation quandle induced by a boundary-parabolic repre…
This study introduces a unified cohomology theory for braided algebras.
problem Classifying infinitesimal deformations of braided algebras.
method Developed a cohomology theory unifying Hochschild and Yang-Baxter cohomology.
result The second cohomology group classifies infinitesimal deformations of braided algebras.
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.
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…
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.
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.