New shadow homotopy invariant defined for links.
problem Homotopy invariants for classical links.
method Defined extended quandle spaces and constructed shadow homotopy invariant.
result Shadow homotopy invariant equals quandle homotopy invariant times quandle order.
The paper defines circular orderability for quandles and explores their properties.
problem Understanding the structure of quandles through circular orderings.
method Introduced circular orderability for quandles, showed embedding properties, and provided examples.
result Spaces of circular orderings embed in the space of all orderings, and examples of non-circularly orderable quandles are given.
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.
A new invariant for links in lens spaces with advantages over traditional methods.
problem Invariants for links in lens spaces.
method Constructing a virtual quandle for links in lens spaces L(p,1). result The virtual quandle is an essential invariant and its presentation can be easily written from a link's band diagram.
Study quandle modules over geometric quandles and their relation to Lie-Yamaguti representations.
problem Understanding quandle modules and their connection to Lie-Yamaguti representations.
method Examine quandle modules over quandle spaces, focusing on geometric structures.
result Modules over quandle spaces are linked to representations of Lie-Yamaguti algebras.
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.
New quandle structures found on topological spaces.
problem No nontrivial quandle structures on [0,1].
method Analyzing topological spaces of positive dimension.
result Infinitely many nonisomorphic quandle structures exist.
Map quandle orders to actions, characterize isolated orders, and prove no isolated right orders.
problem Characterizing isolated orders on quandles and proving the absence of certain orders.
method Construct a continuous map from orders to actions, use strong rigidity, and analyze specific cases.
result No isolated right orders on free quandles, except for specific cases.
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. 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.
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 Riemannian symmetric spaces, and classify flat connected finite quandles.
Enhances knot invariants using bilinear forms on vector spaces.
problem Improving classical and virtual knot invariants.
method Uses bilinear forms on vector spaces indexed by pairs of elements of a finite quandle.
result New enhanced invariants of knots and links.
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. We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the symplectic space Z⊕Z. The last quandle can be identified with the Dehn quandle of the torus and the cord quandle on a 2-sphere with four punctures. We also show that the fund…
We introduce a notion of topological quandle. Given a topological quandle Q we associate to every classical link L in R3 an invariant JQ(L) which is a topological space (defined up to a homeomorphism). The space JQ(L) can be interpreted as a space of colourings of a diagram of the link L with colours f…
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.
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.
Study satellite knots and their quandles related to incompressible tori.
problem Understanding quandles of satellite knots and their components.
method Algebraic approach to augmented fundamental quandles, presentations of fundamental quandles, and analysis of Alexander modules.
result Relationships between satellite knots, companion and pattern knots, and their fundamental quandles.
New shifting chain map enhances quandle invariants for links.
problem Enhancing quandle invariants for links and surfaces.
method Introducing a shifting chain map σ and its pull-back σ# to transform cocycles.
result Shifting chain map σ# transforms 2-cocycles to 3-cocycles, enhancing invariants.
Study de Rham theory for cubical manifolds and quandles.
problem Rational homotopy type of classifying spaces of smooth quandles.
method Show de Rham theory for cubical manifolds and study secondary characteristic classes.
result Secondary characteristic classes produce cocycles of quandles.
Quandles can be regarded as generalizations of symmetric spaces. Among symmetric spaces, two-point homogeneous Riemannian manifolds would be the most fundamental ones. In this paper, we define two-point homogeneous quandles analogously, and classify those with prime cardinality.
Paper introduces tensor product of quandles for knot classification.
problem Classifying knot invariants of surface-links with 1-handles.
method Introduces tensor product of quandles and applies to surface-links.
result Tensor product of knot quandles/classical quandles can classify surface-link invariants.
The paper studies quandles from symmetric group automorphisms and finds a correspondence with conjugacy classes.
problem Investigating quandles from automorphisms of symmetric groups.
method Developed quandle invariants and proved a one-to-one correspondence between quandles and conjugacy classes.
result There is a one-to-one correspondence between generalized Alexander quandles arising from symmetric groups $\Sf_n$ and the conjugacy classes of $\Sf_n$ for 3≤n≤30 with neq6,15. This paper constructs quandles with abelian inner automorphism groups from graphs, proving their homogeneity.
problem Finding quandles with specific automorphism properties.
method Starting from simple graphs, the paper constructs quandles with abelian inner automorphism groups and proves their homogeneity.
result Homogeneous quandles with abelian inner automorphism groups are constructed from vertex-transitive graphs.
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…
New algebraic structures for topological pairs.
problem No specific problem stated; generalization of knot quandles.
method Introducing multi-quandles for topological pairs.
result New algebraic structures for topological pairs.
We define a quandle variety as an irreducible algebraic variety Q endowed with an algebraically defined quandle operation ⊳. It can also be seen as an analogue of a generalized affine symmetric space or a regular s-manifold in algebraic geometry. Assume that Q is normal as an algebraic variety and that the a…
Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.
problem Understanding the homotopy types of free racks and quandles.
method Proved analogs of Milnor's theorem for racks and quandles and their pointed variants.
result Identified the homotopy types of free racks and quandles on spaces of generators.
This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles X, from a perspective of homotopy theory: Specifically, for any prime ℓ which does not divide the type of X, the ℓ-torsion of this invariants is equal to a sum of the colouring po…
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.
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.
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.
Paper computes knot symmetric quandle for surface-links and finds infinitely many distinct surface-knots.
problem Computing knot symmetric quandle for surface-links.
method Using plat form presentations, the paper computes the knot symmetric quandle for surface-links.
result Infinitely many distinct surface-knots of genus g with plat indices m.
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 introduces derivations for quandles and their properties.
problem Defining and studying derivations for quandles.
method Introducing derivations as twisted analogues of quandle homomorphisms and proving properties of derived quandles.
result For each quandle Q, there exists a unique derived quandle AQ with specific properties. 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 their third homologies. For example, all Mochizuki's quandle 3-cocycles are sho…
Free and knot quandles are residually finite.
problem Residual finiteness of quandles.
method Definition and investigation of residual finiteness; proof for free and knot quandles; discussion on automorphism groups.
result Free and knot quandles are residually finite and Hopfian.
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 field invariant refines real spectrum and relates to absolute Galois group.
problem Understanding field invariants related to absolute Galois groups.
method Introducing Artin-Schreier quandles and computing their properties for different types of fields.
result Artin-Schreier quandles provide relations between fields and their absolute Galois groups.
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.
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 …