The paper extends a result about involutory medial quandles to links, providing bounds and stronger invariants.
problem Extending the involutory medial quandle concept to links and understanding its properties.
method Analyzing the involutory medial quandle of classical links and comparing it to the homology group.
result The order of the involutory medial quandle is bounded and can be infinite only if the determinant is zero.
The study shows involutory quandles of certain links are not left-orderable.
problem Determining left-orderability of involutory quandles of links.
method Using a non-left-orderability criterion for involutory quandles of non-split links, the study improved previous results and introduced new families of links.
result The involutory quandles of non-trivial alternating links and certain augmented alternating links are not left-orderable.
The paper extends Alexander invariant and medial quandle equivalence to links.
problem Understanding stronger invariants for links than Alexander modules.
method Extending Joyce's observation to multiple medial quandles and reduced Alexander modules.
result Medial quandles provide stronger invariants than reduced Alexander modules for 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.
We present an overview of some older papers on involutory quandles, mostly from the times before the term "quandle" was born. It is meant as a reference guide, not (yet) as an expository article explaining what the involutory quandles are and what they are good for.
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.
Medial quandles are represented using a heterogeneous affine structure. As a consequence, we obtain numerous structural properties, including enumeration of isomorphism classes of medial quandles up to 13 elements.
Study of Hom quandles, focusing on structure and counting.
problem Characterizing the structure of Hom quandles.
method Analyzing quandles of homomorphisms into medial quandles, focusing on medial source quandles.
result Characterization of Hom quandles, especially when the target is 2-reductive.
A quandle orbit's orientation is problematic when reversed.
problem The natural orientation-reversal of quandle orbits is unsuitable for medial quandles.
method Defined the orientation-reversal of a quandle orbit by inverting translations, observed it's unsuitable for medial quandles.
result The natural orientation-reversal of quandle orbits is unsuitable for medial quandles.
In this paper we show that Montesinos links of the form L(1/2, 1/2, p/q;e), which we call (2,2,r)-Montesinos links, have finite involutory quandles. This generalizes an observation of Winker regarding the (2, 2, q)-pretzel links. We also describe some properties of these quandles.
Medial quandles fail to distinguish certain links, highlighting their limitations.
problem Detecting causality in spacetime using quandles.
method Investigated medial quandles' ability to distinguish specific links and knots.
result Medial quandles fail to distinguish certain links, including the connected sum of two Hopf links from an infinite series of relevant three-component links.
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.
A quandle is a self-distributive algebraic structure that appears in quasi-group and knot theories. For each abelian group A and c \in A we define a quandle G(A, c) on \Z_3 \times A. These quandles are generalizations of a class of non-medial Latin quandles defined by V. M. Galkin so we call them Galkin quandles. Each …
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.
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.
Enhances quandle and biquandle colorings using biquandle structures.
problem Improving quandle and biquandle colorings.
method Investigates biquandle structures and biquandle homomorphisms.
result Describes biquandle structure of the Hom-biquandle and relates Hom-quandles to Hom-biquandles.
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. 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 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. Study of generalized Legendrian racks and their GL-structures.
problem Characterizing and classifying generalized Legendrian racks.
method Algebraic analysis, classification of racks and GL-racks, tensor products.
result Classification of several infinite families of GL-racks.
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.
This paper develops graph theory for racks and quasigroups.
problem Characterizing and realizing right quasigroups and related structures.
method Study of graph markings, Schreier graphs, and Cayley graphs.
result All right quasigroups are realizable by specific types of graphs.
The involutory birack counting invariant is an integer-valued invariant of unoriented tangles defined by counting homomorphisms from the fundamental involutory birack of the tangle to a finite involutory birack over a set of framings modulo the birack rank of the labeling birack. In this first of an anticipated series …
We identify a subcategory of biracks which define counting invariants of unoriented links, which we call involutory biracks. In particular, involutory biracks of birack rank N=1 are biquandles, which we call bikei. We define counting invariants of unoriented classical and virtual links using finite involutory biracks, …
We consider involutory virtual biracks with good involutions, also known as symmetric involutory virtual biracks. Any good involution on an involutory virtual birack defines an enhancement of the counting invariant. We provide examples demonstrating that the enhancement is stronger than the unenhanced counting invarian…
Examines medial axis in pseudo-Euclidean spaces.
problem No specific problem stated; focuses on new context.
method Follows Birbrair and Denkowski's approach.
result Feasibility of medial axis in pseudo-Euclidean spaces checked.
We consider the question of which virtual knots have finite fundamental medial bikei. We describe and implement an algorithm for completing a presentation matrix of a medial bikei to an operation table, determining both the cardinality and isomorphism class of the fundamental medial bikei, each of which are link invari…
In the 90s, based on presentations of 3-manifolds by Heegaard diagrams, Kuperberg associated a scalar invariant of 3-manifolds to each finite dimensional involutory Hopf algebra over a field. We generalize this construction to the case of involutory Hopf algebras in arbitrary symmetric monoidal categories admitting cer…
The Blum medial axis rigidity is studied in terms of cross ratios and differential geometry.
problem Examining rigidity properties of the Blum medial axis under diffeomorphisms.
method Using cross ratios from projective geometry and differential geometry.
result Cross ratios and differential geometry uniquely determine the angles between smooth sheets.
Introduces entropic tribrackets and their applications in link distinguishing.
problem Distinguishing links with the same counting invariant.
method Definition and study of entropic tribrackets and their homsets.
result Homsets of entropic tribrackets form new entropic tribrackets.
This paper consists of two parts. In the first one we study the behaviour of medial axes (skeletons) of closed sets in a connected complete Riemannian manifold M under deformations. The second one is devoted to a similar study of conflict sets. We apply a new approach to the deformation process. Instead of …
Combining M-algebra and hyperbolic involutory algebra extends exceptional tangent spaces to 11 dimensions.
problem Combining symmetries in M-theory to extend exceptional tangent spaces.
method Combining known results to show hyperbolic involutory algebra acts on M-algebra through brane-rotating symmetry.
result Extends the hierarchy of exceptional tangent spaces from n ≤ 7 to n = 11.
Involutory Hopf group-coalgebras provide new invariants for 4-manifold bundles.
problem Developing invariants for flat bundles over 4-manifolds.
method Utilizing Hopf G-triplets and colored trisection diagrams. result Involutory Hopf G-triplets yield well-defined invariants of G-colored trisection diagrams. The study classifies graphs on surfaces with positive curvature properties.
problem Classifying graphs on surfaces with specific curvature properties.
method Using medial graphs and classification techniques.
result Complete classification of graphs on surfaces with positive Forman curvature and corner curvature.
In this paper we study a robust expected utility maximization problem with random endowment in discrete time. We give conditions under which an optimal strategy exists and derive a dual representation for the optimal utility. Our approach is based on a general representation result for monotone convex functionals, a fu…
Invariants for 4-manifolds from Hopf group-algebras.
problem Constructing invariants for flat connections on 4-manifolds.
method Using finite type involutory quasitriangular Hopf G-algebras and coloring Kirby diagrams. result Invariants defined for 4-manifolds and connections.
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.
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 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.
Invariants for 3-manifolds with embedded links using Hopf algebras.
problem Constructing invariants for 3-manifolds with embedded framed links.
method Using involutory Hopf algebras and representations of Drinfeld double.
result The invariant recovers known invariants for specific cases.
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.