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.
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.
We define a knot/link invariant using set theoretical solutions (X,σ) of the Yang-Baxter equation and non commutative 2-cocycles. We also define, for a given (X,σ), a universal group Unc(X) governing all 2-cocycles in X, and we exhibit examples of computations.
Criterion for lifting smooth contact maps between Carnot groups to central extensions.
problem Existence of smooth contact map lifts between Carnot groups and their central extensions.
method Criterion using pullbacks and Lie algebra cohomology classes.
result Necessary and sufficient conditions for lifting are formulated.
New knot invariants derived from non-abelian Yang-Baxter solutions.
problem Developing new invariants for virtual and classical knots/links.
method Defining invariants using non-commutative 2-cocycles pairs and groups.
result Examples of computations achieved using GAP.
New biquandle bracket invariants are linked to biquandle 2-cocycles.
problem Quantum enhancements and biquandle colored links.
method Proving biquandle bracket invariants are pointwise products of other invariants and biquandle 2-cocycles.
result New biquandle bracket invariants are equivalent to the Jones polynomial on knots.
The paper introduces biquandle (co)homology and handles invariants for handlebody-links.
problem Developing invariants for handlebody-links using biquandle (co)homology.
method Introducing biquandle (co)homology and constructing invariants using 2-cocycles.
result Constructs invariants for S^1-oriented handlebody-links using 2-cocycles.
Derives equations for interacting Lie-Poisson systems using 2-cocycle extensions.
problem Understanding collective motion of interacting Lie-Poisson systems.
method Derives equations on dual space of extended structure, including 2-cocycle terms.
result Provides most general realization of Lie-Poisson system coupling.
Adjusts Yang-Baxter operators for HOMFLYPT polynomials.
problem Computing 2-cocycle invariants for links.
method Adjusts Yang-Baxter operators and computes second homology.
result Potential for 2-cocycle invariant for links.
For an arbitrary identity L=R between compositions of maps L and R on tensors of vector spaces V, a general construction of a 2-cocycle condition is given. These 2-cocycles correspond to those obtained in deformation theories of algebras. The construction is applied to a canceling pairings and copairings, with explicit…
Continuous cohomology theory for topological quandles introduced and compared.
problem Developing a continuous cohomology theory for topological quandles.
method Introduced continuous cohomology theory, compared to algebraic theories, studied extensions with continuous 2-cocycles, computed cohomology groups of inverse limits.
result Showed differences in second cohomology groups for specific topological quandles.
Extended current algebra on S^3 with new bilinear form and 2-cocycle.
problem Investigate the current algebra on S^3 of complex Lie algebras.
method Defined a new quaternion-valued 2-cocycle and symmetric invariant bilinear form.
result Extended affine Kac-Moody algebra to Lie algebra of smooth mappings.
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. New 2-cocycles introduced on Lie algebra of S^3H, leading to central extensions and root space decompositions.
problem Introducing new 2-cocycles on Lie algebra of S^3H and their implications.
method Extending 2-cocycles to Lie algebra S3gl(n,H), defining central extensions and root spaces. result Obtained root space decomposition and Chevalley generators for the Lie algebra sl^(n,H). Infinite group knot coloring polynomial generalizes quandle 2-cocycle invariant.
problem Generalizing knot invariants to infinite groups.
method Longitudinal mapping invariant based on meridian-longitude pair in knot group.
result Invariant values for specific knots and groups.
The paper studies metrics on Lie groups SO(2) and SO(3) and their integrability.
problem Integrability of gradient systems on Lie groups via Fisher metrics.
method Analysis of Souriau-Fisher metrics and 2-cocycles on Lie groups SO(2) and SO(3).
result Cocycles can locally modify Fisher metrics on Lie group orbits.
Study on flux homomorphism and its extension in symplectic group of a disk.
problem Understanding the flux homomorphism and its extension in symplectic group.
method Defined and analyzed the flux homomorphism and its extension, determined the Euler class, and investigated its relation to group 2-cocycle and Calabi invariant.
result Determined the Euler class of the flux extension and investigated its relation to group 2-cocycle and Calabi invariant.
Study invariants of Z/p-homology 3-spheres from abelianization of mapping class groups.
problem Deciding and constructing invariants of Z/p-homology 3-spheres. method Formulating a criterion and using families of trivial 2-cocycles on the abelianization of the level-p mapping class group. result Disproved a conjectured extension of the Casson invariant for rational homology 3-spheres.
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.
In this note we prove that for any finite quandle X and any 2-cocycle $φ\in Z^2_{Q\pm}(X; \mathds{Z})$, the cocycle invariant Φφ±(K) is trivial for all knots K.
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.
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.
We introduce the notion of quasi-triviality of quandles and define homology of quasi-trivial quandles. Quandle cocycle invariants are invariant under link-homotopy if they are associated with 2-cocycles of quasi-trivial quandles. We thus obtain a lot of numerical link-homotopy invariants.
Researchers attempt to categorify biquandle brackets using Khovanov homology methods.
problem Categorify biquandle brackets using Khovanov homology methods.
method Outline a Khovanov homology-style construction for biquandle brackets.
result A canonical biquandle 2-cocycle is defined, but not a true categorification of biquandle brackets.
We define families of invariants for elements of the mapping class group of S, a compact orientable surface. Fix any characteristic subgroup H of pi_1(S) and restrict to J(H), any subgroup of mapping classes that induce the identity modulo H. To any unitary representation, r of pi_1(S)/H we associate a higher-order rho…
Modified bikei homology enhances knot and surface invariants.
problem Enhancing knot and surface invariants using bikei.
method Using bikei 2-cocycles to modify homology and cohomology.
result Improved counting invariant for knots and surfaces.
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.
In this paper, we provide a construction of a state-sum model for finite gauge-group Dijkgraaf-Witten theory on surfaces with codimension 1 defects. The construction requires not only that the triangulation be subordinate to the filtration, but flag-like: each simplex of the triangulation is either disjoint from the de…
The jet bundle JkG of k-jets of curves in a Lie group G has a natural Lie group structure. We present an explicit formula for the group multiplication in the right trivialization and for the group 2-cocycle describing the abelian Lie group extension g→JkG→Jk−1G.
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…
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…
The paper studies a special Grassmannian space and shows it's an orbit of a unitary group.
problem Investigating a specific Grassmannian space of infinite-dimensional subspaces.
method Analyzing the restricted p-Schatten class Grassmannian and showing it's an affine coadjoint orbit of a unitary group. result The restricted p-Schatten class Grassmannian is shown to be an affine coadjoint orbit of an infinite-dimensional restricted unitary group. 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…
We define counting and cocycle enhancement invariants of virtual knots using parity biquandles. These invariants are determined by pairs consisting of a biquandle 2-cocycle φ^0 and a map φ^1 with certain compatibility conditions leading to one-variable or two-variable polynomial invariants of virtual knots. We provide …
Paper proves Rohlin invariant's uniqueness and extends homology sphere invariants.
problem Proving the uniqueness of Rohlin invariant and extending homology sphere invariants.
method Using the Rohlin invariant's uniqueness, the paper extends invariants from trivial 2-cocycles to those with 2-torsion.
result Generalized invariants of homology spheres with 2-torsion values.
New polynomial invariants derived from birack and switch structures.
problem Polynomial invariants of braids.
method Switch structures, birack colorings, quiver-valued invariants.
result New polynomial invariants of braids.
New bialgebra structures for relative Poisson algebras are introduced.
problem Extending bialgebra structures from commutative differential algebras to relative Poisson algebras.
method Introducing new bialgebra structures (relative PCA bialgebras) and using commutative 2-cocycles.
result New bialgebra structures (relative PCA bialgebras) are equivalent to certain Manin triples.
New invariants for link analysis include biquandle power brackets.
problem Analyzing oriented links with new invariants.
method Introducing biquandle power brackets as an infinite family of link invariants.
result Biquandle power brackets encompass classical and previous invariants.
We construct eta- and rho-invariants for Dirac operators, on the universal covering of a closed manifold, that are invariant under the projective action associated to a 2-cocycle of the fundamental group. We prove an Atiyah-Patodi-Singer index theorem in this setting, as well as its higher generalization. Applications …
We prove that if G is an abelian group of odd order then there is an isomorphism from the second quandle homology of the Takasaki quandle of G to the exterior square of G. In particular, for G=Z_k^n, k odd, we obtain Z_k^{n(n-1)/2}. Nontrivial second homology allows us to use 2-cocycles to construct new quandles from T…
This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let π_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longit…
In CJKLS quandle cohomology is used to produce invariants for particular embeddings of codimension two; 2-cocycles give to invariants for (classical) knots and 3-cocycles give rise to invariants for knotted surfaces. This is done by way of a notion of coloring of a diagram. Also, these invariants have the form of state…
New invariants defined for knots and links using quandle representations.
problem Defining new invariants for knots and links.
method Defined a family of quiver representations associated to finite quandles, abelian groups, and quandle 2-cocycles.
result Computed four new polynomial invariants for knots and links.
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…
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.
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.
New topological Riemann-Roch theorem for circle fibrations.
problem Topological Riemann-Roch theorem for complex line bundles on circle fibrations.
method Construction of central extensions and application to algebraic K-theory.
result Equality of specific cohomology elements in the third cohomology group.
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.