New rack and multiple group rack cohomology for surfaces in 3-sphere.
problem Categorizing compact oriented surfaces in 3-sphere based on symmetry.
method Developed cohomology theory for racks and multiple group racks, constructed cocycle invariants.
result Identified new symmetry types of surfaces in 3-sphere.
New method to calculate 3-manifold invariants via skew-racks.
problem Calculating invariants of 3-manifolds.
method Introducing skew-racks with good involution and Property FR, defining cocycle invariants.
result Established new approach to obtain 3-manifold invariants via Dehn surgery.
This paper characterizes extensions of augmented racks and constructs invariants for surfaces.
problem Characterizing extensions of augmented racks and constructing invariants for surfaces.
method Characterization of rack extensions through fibrant and additive cohomology, construction of invariants using cocycles.
result Characterization of extensions of augmented racks and construction of surface invariants.
We introduce the notion of N-reduced dynamical cocycles and use these objects to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide examples to show that the new invariants are not determined by the rack counting invariant, the Jones polynomial or the generalized Al…
The aim of this paper is to define a homology theory for racks with finite rank N and use it to define invariants of knots generalizing the CJKLS 2-cocycle invariants related to the invariants defined in [15]. For this purpose, we prove that N -degenerate chains form a sub-complex of the classical complex defining rack…
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…
The homology and cohomology of quandles and racks are used in knot theory: given a finite quandle and a cocycle, we can construct a knot invariant. This is a quick introductory survey to the invariants of knots derived from quandles and racks.
Link invariants defined from finite crossed modules and Reidemeister pairs.
problem Defining link invariants from finite categorical groups.
method Definition of tangle and framed tangle invariants using finite crossed modules and Reidemeister pairs.
result Includes all rack and quandle cohomology (framed) link invariants and the Eisermann invariant of knots.
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, 2-crossed modules and braided crossed modules. We…
Heap theory applied to framed links yields new invariants.
problem Developing invariants for framed links using heap theory.
method Introducing fundamental heap, defining cocycle invariant using ternary cohomology.
result Found cocycles and computed invariants for specific link families.
We introduce augmented biracks and define a (co)homology theory associated to augmented biracks. The new homology theory extends the previously studied Yang-Baxter homology with a combinatorial formulation for the boundary map and specializes to N-reduced rack homology when the birack is a rack. We introduce augmente…
Quandle identities give rise to subcomplexes in homology.
problem Understanding the relationship between quandle identities and their homological counterparts.
method Examining subcomplexes constructed from quandle identities and their invariance under Reidemeister moves.
result Quandle identities give rise to 2-cycles in homology, and these cycles can be extended to form abelian extensions.
Zesting affects Reshetikhin-Turaev invariants of links and 3-manifolds.
problem Understanding how zesting affects Reshetikhin-Turaev invariants.
method Developed a local formalism to compute tangle invariants and link invariants.
result Zesting contributes to complexity-theoretic hierarchies of topological field theories.
The paper connects GL-racks to knot coloring invariants.
problem Understanding invariants of Legendrian knots.
method Exploring GL-racks and their decomposition into permutation and block GL-racks.
result Equivalent coloring invariants for knots with identical classical invariants.
This article gives a local answer to the coquecigrue problem. Hereby we mean the problem, formulated by J-L. Loday in \cite{LodayEns}, is that of finding a generalization of the Lie's third theorem for Leibniz algebra. That is, we search a manifold provided with an algebraic structure which generalizes the structure of…
We study Coxeter racks over Zn and the knot and link invariants they define. We exploit the module structure of these racks to enhance the rack counting invariants and give examples showing that these enhanced invariants are stronger than the unenhanced rack counting invariants.
Study rack invariants for links in lens space L(p,1).
problem Applying classical rack invariants to links in L(p,1). method Presented augmented fundamental rack, applied counting rack invariants, included information about π1(L(p,1)) action. result Counting rack invariants provide information about π1(L(p,1)) action on links in L(p,1). A rack shadow is a set X with a rack action by a rack R, analogous to a vector space over a field. We use shadow colorings of classical link diagrams to define enhanced rack counting invariants and show that the enhanced invariants are stronger than unenhanced counting invariants.
New algebraic structure helps distinguish braids.
problem Distinguishing braids using mathematical invariants.
method Defined pointed racks and used them to create braiding invariants.
result New invariants can distinguish braids not previously possible.
We introduce a modified rack algebra Z[X] for racks X with finite rack rank N. We use representations of Z[X] into rings, known as rack modules, to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide computations and examples to show that the new invariants are stric…
We study rack polynomials and the link invariants they define. We show that constant action racks are classified by their generalized rack polynomials and show that nsata-quandles are not classified by their generalized quandle polynomials. We use subrack polynomials to define enhanced rack counting invariants, gen…
New rack invariants detect geometric properties of Legendrian knots.
problem Detecting geometric properties of Legendrian knots.
method Introducing Legendrian racks, a generalization of quandle invariants.
result These invariants form a natural generalization of quandle invariants.
New method constructs multiple group racks, differing from known constructions.
problem Define new invariants for spatial surfaces.
method Using a G-family of racks and a normal subgroup N of G.
result New method yields multiple group racks not derived from known methods.
A (t,s)-rack is a rack structure defined on a module over the ring Λ¨=Z[t±1,s]/(s2−(1−t)s). We identify necessary and sufficient conditions for two (t,s)-racks to be isomorphic. We define enhancements of the rack counting invariant using the structure of (t,s)-racks and give some computations an…
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.
New algebraic structures help distinguish Legendrian knots.
problem Distinguishing Legendrian knots from smooth knots.
method Defined Legendrian racks and used them to define invariants.
result Distinguished certain Legendrian knots.
4-Legendrian permutation racks can't distinguish knots but recover classical invariants.
problem Distinguishing Legendrian knots using permutation racks.
method Study of 4-Legendrian racks and their effectiveness.
result 4-Legendrian permutation racks cannot distinguish knots but recover classical invariants.
Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then it admits no rack colorings. In this paper, we investigate rack colorings for surface-knot diagrams without branch points and prove that rac…
Innovative rack theory applied to Legendrian links.
problem Classifying and distinguishing Legendrian links.
method Purely rack-theoretic approach, Legendrian Reidemeister moves, cusps, homogeneous representations, modules.
result Invariant distinguishes infinitely many Legendrian unknots and trefoils.
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.
Paper proves Legendrian knots with same GL-rack have similar invariants.
problem Classifying Legendrian knots based on their invariants.
method Examined fundamental GL-racks and their relationship to Thurston-Bennequin and rotation numbers.
result Two Legendrian knots with isomorphic fundamental GL-racks have similar invariants.
A birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In this paper we extend the counting invariant for finite racks to the case of finite biracks. We introduce a family of biracks generalizi…
The paper introduces groupoid racks for spatial surfaces.
problem Coloring diagrams of spatial surfaces for invariant calculation.
method Introduces groupoid racks with universal properties.
result Groupoid racks provide an invariant for spatial surfaces.
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 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…
Analytic Lie rack structures on Leibniz algebras are characterized and rigid Lie algebras are identified.
problem Characterizing and identifying rigid Lie algebras with analytic Lie rack structures.
method Analytic Lie rack structures are defined and characterized using multilinear equations and cohomological interpretations.
result Simple Lie algebras are conjectured to be rigid as left Leibniz algebras.
Extends Borel invariant to measurable cocycles of 3-manifold groups.
problem Defining and analyzing Borel invariant for measurable cocycles.
method Introducing pullback along measurable cocycles and extending Borel invariant.
result Maximal cocycles are trivializable to irreducible representations.
New (co)homology theory for symmetric quandles developed.
problem Developing strong invariants for symmetric quandles.
method Introducing symmetric quandle modules and Beck modules, extending module theory, and constructing generalized (co)homology.
result Established an explicit isomorphism between symmetric quandle cohomology and group cohomology.
New graph-based invariants from quandle cocycles.
problem Defining new link invariants from quandle cocycles.
method Integrating quandle cocycle information into quandle coloring quivers to create weighted directed graphs.
result Definition of new link invariants including a 2-variable polynomial.
Explicit adjoint group description for Coxeter quandles.
problem Understanding the adjoint group structure of Coxeter quandles.
method Explicit descriptions and constructions of adjoint groups, using central extensions and 2-cocycles.
result The adjoint group of a Coxeter quandle is an intermediate group between the Coxeter group and its Artin group, with specific properties related to commutator subgroups and root systems.
New knot invariants computed without explicit cocycles.
problem Computing knot invariants without explicitly finding cocycles.
method Using generalized Alexander quandles and colorings of 1-tangles.
result The 2-cocycle invariant distinguishes many prime knots.
T. Mochizuki determined all 3-cocycles of the third quandle cohomologies of Alexander quandles on finite fields. We show that all the 3-cocycles, except those of 2-cocycle forms, are derived from group 3-cocycles of a meta-abelian group. Further, the quandle cocycle invariant of a link using Mochizuki's 3-cocycle is eq…
Virtual index cocycles reformulate virtual link invariants.
problem No specific problem stated; focuses on reformulation.
method Using virtual index cocycles to reformulate invariants.
result Unified reformulation of virtual link invariants.
Paper studies quandle shadow cocycle invariants and Vassiliev invariants.
problem Relationship between quandle shadow cocycle invariants and Vassiliev invariants.
method Proves that the coefficient of the finite perturbative expansion of the quandle shadow cocycle invariant is a Vassiliev invariant for any braids.
result Coefficient of quandle shadow cocycle invariant is a Vassiliev invariant.
Enhances psyquandle counting invariants using cocycles.
problem Improving the counting invariant of pseudoknots and singular knots.
method Defining enhancements via biquandle 2-cocycles and new functions.
result New polynomial invariants that are proper enhancements and not determined by the Jablan polynomial.
Quantum cocycle invariants derived from Yang-Baxter cohomology.
problem Constructing stronger quantum knot invariants.
method Developing quantum cocycle invariants using Yang-Baxter cohomology and deformation theory.
result Quantum cocycle invariants yield stronger invariants in certain examples.
We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virt…
Quandle 2-cocycles yield invariant values for knots under certain algebraic conditions.
problem Defining and understanding invariants of knots using quandle 2-cocycles.
method Analyzing algebraic properties of quandle extensions and their impact on knot invariants.
result The invariant values are constant or follow a restricted form for classical knots under specific conditions.