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.
Graph morphism maps Poisson cocycles to symmetries, revealing factorization through Jacobi identity.
problem Mapping graph cocycles to symmetries of Poisson structures.
method Kontsevich graph orientation morphism and differential consequences of Jacobi identity.
result Existence of factorization through differential consequences of Jacobi identity.
The paper constructs non-trivial cocycles for long embeddings with more than one loop.
problem Constructing non-trivial cocycles for long embeddings with more than one loop.
method Integral over configuration spaces associated with Bott-Cattaneo-Rossi graphs with more than one loop.
result Explicit construction of a non-trivial family of trivial long embeddings for odd dimensions.
By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken …
Constructs integer-valued cohomology classes from graph cocycles.
problem Constructing nontrivial cohomology classes from graph cocycles.
method Gluing compactified configuration spaces to construct integer-valued classes from integer-valued graph cocycles.
result Obtains nontrivial classes from trivalent graph cocycles.
Graph complex acts on Poisson bi-vectors, producing universal cocycles.
problem Understanding the action of graph complex on Poisson bi-vectors.
method Using Lie derivatives and graph cocycles, the graph complex acts on Poisson bi-vectors.
result A uniform construction of universal cocycles for homogeneous Poisson bi-vectors.
Paper discusses biquandle cohomology and invariants for surface-links.
problem Computing invariants for surface-links using biquandle theory.
method Developed (co)homology theory of biquandles, introduced new invariants using broken surface diagrams and marked graph diagrams.
result Constructs new invariants for surface-links using biquandle theory.
In this paper we show that via the configuration space integral construction a non-trivalent graph cocycle can also yield a non-zero cohomology class of the space of higher (and even) codimensional long knots. This simultaneously proves that the Browder operation induced by the operad action defined by R. Budney is not…
By 2-twist-spinning the knotted graph that represents the knotted handlebody 52, we obtain a knotted foam in 4-dimensional space with a non-trivial quandle cocycle invariant.
We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.
New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.
problem Defining invariants for framed 3-manifolds with semi-simple Lie groups.
method Constructing graph complexes and cocycles, including self-loops, to define invariants.
result Higher-loop invariants can be defined by graph cocycles with or without self-loops.
Study cohomology of GL₂n(Z) and graph complexes using Pfaffian forms.
problem Cohomology of GL₂n(Z) and related graph complexes.
method Use Pfaffian forms on symmetric spaces and dual Laplacians on graphs.
result First cocycle gives a non-trivial class in H⁻⁶(GC₃).
The paper proves rigidity for cocycles from higher rank lattices to Out(FN).
problem Proving rigidity for cocycles from higher rank lattices to Out(FN).
method Geometric tool: barycenter map.
result Every Borel cocycle is cohomologous to a cocycle with finite image.
Introduces fundamental heaps for surfaces, linking them to cocycle invariants.
problem Isotopy invariants of surface embeddings in 3-space.
method Definition of fundamental heaps using surface ribbons and heap operations.
result Fundamental heaps have a free part whose rank matches the number of connected components.
New Vassiliev invariant of order three derived from Fox-Hatcher cycles.
problem Developing a new Vassiliev invariant of order three.
method Integration of a 1-cocycle over Fox-Hatcher 1-cycles.
result Derives a Vassiliev invariant of order three.
In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued co…
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.
We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We…
Study projective derivative cocycles for circle diffeomorphisms.
problem Understanding reducibility and almost reducibility in circle diffeomorphisms.
method Computing precise expressions for projective derivative cocycles and extending to 3-torus.
result Generalization of results to diagonal action on 3-torus.
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.
We give a construction of quandle cocycles from group cocycles, especially, for any integer p \geq 3, quandle cocycles of the dihedral quandle R_p from group cocycles of the cyclic group Z/p. We will show that a group 3-cocycle of Z/p gives rise to a non-trivial quandle 3-cocycle of R_p. When p is an odd prime, since d…
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.
Study on group cocycles for volume-preserving diffeomorphisms.
problem Understanding group cocycles on volume-preserving diffeomorphisms.
method Constructed two types of group cocycles on the volume-preserving diffeomorphism group.
result One cocycle yields the Euler class of flat sphere bundles for the sphere.
The paper extends Euler class theory to measurable cocycles.
problem Understanding the structure of measurable cocycles and their cohomology.
method Constructing a parametrized Euler class in bounded cohomology and studying semicohomologous cocycles.
result The parametrized Euler class vanishes if and only if the cocycle can be lifted and admits an equivariant family of points.
We derive general Novikov-Morse type inequalities in a Conley type framework for flows carrying cocycles, therefore generalizing our results in [FJ2] derived for integral cocycle. The condition of carrying a cocycle expresses the nontriviality of integrals of that cocycle on flow lines. Gradient-like flows are distingu…
We study the hairy graph homology of a cyclic operad; in particular we show how to assemble corresponding hairy graph cohomology classes to form cocycles for ordinary graph homology, as defined by Kontsevich. We identify the part of hairy graph homology coming from graphs with cyclic fundamental group as the dihedral h…
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.
Researchers compute a residue cocycle for Dirac-type operators using modified Getzler calculus.
problem Computing the residue cocycle for Dirac-type operators.
method Modified Getzler calculus for computation.
result Computed residue cocycle for a class of Dirac-type operators.
New algebraic rules for 5D shapes based on 3D cocycles.
problem Creating rules for 5D shapes.
method Using simplicial 3-cocycles to parameterize heptagon relations.
result Parameterized heptagon relations for 5D shapes.
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…
We consider cocycles of isometries on spaces of nonpositive curvature H. We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there…
The theory of quandle (co)homology and cocycle knot invariants is rapidly being developed. We begin with a summary of these recent advances. One such advance is the notion of a dynamical cocycle. We show how dynamical cocycles can be used to color knotted surfaces that are obtained from classical knots by twist-spinnin…
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.
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.
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.
Develops Patterson-Sullivan theory for coarse cocycles.
problem None explicitly stated in the abstract.
method Theory of Patterson--Sullivan measures for coarse cocycles of convergence groups.
result Existence, uniqueness, and ergodicity results for Patterson-Sullivan measures under geometric assumptions.
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.
Paper extends multiplicative constants to measurable cocycles theory.
problem Maximal measurable cocycles in bounded cohomology.
method Extending multiplicative constants to measurable cocycles theory.
result Defined and studied Cartan invariant for measurable PU(m,1)-cocycles.
Study of Penner's cocycle on fatgraph complex.
problem Understanding Penner's cocycle on fatgraph complex.
method Explicit cobounding cochain formula involving summation over trivalent vertices.
result Expressed symplectic form of surface underlying fatgraph.
Computes algebraic hull of Kontsevich-Zorich cocycle.
problem Finiteness of GL^+_2(R) invariant subvarieties of the Hodge bundle.
method Computes algebraic hull over GL^+_2(R) invariant subvarieties.
result Finiteness results on invariant subvarieties.
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.
New invariant for spin 3-manifolds using super 3-cocycles.
problem Constructing an invariant for spin 3-manifolds.
method State sum construction based on super 3-cocycles and combinatorial representation.
result The invariant is sensitive to the spin structure.
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.
The paper extends cyclic cocycles and proves relative index theorems for partitioned manifolds.
problem Index theorems on partitioned manifolds.
method Extending Roe's cyclic 1-cocycle to relative settings and proving index theorems using the cyclic cocycle.
result Generalizations of index theorems on partitioned manifolds.
Paper studies knotoid chirality using shadow quandle colorings and invariants.
problem Distinguishing knotoids from their mirrors.
method Shadow quandle colorings and cocycle invariants.
result Knotoid 31 is shown to be chiral. Diffeomorphism cocycles over hyperbolic systems are bounded and isometric.
problem Understanding the boundedness and isometry of diffeomorphism cocycles.
method Analyzing periodic data and using Hölder continuity.
result Cocycles with bounded periodic data are isometric with respect to Hölder metrics.
Paper shows equivalences between Dixmier-Douady bundles and differential 3-cocycles.
problem Understanding equivalences between Dixmier-Douady bundles and differential 3-cocycles.
method Focuses on the model of Dixmier-Douady bundles and provides equivalences between 2-stacks.
result Equivalence between Dixmier-Douady bundles and differential 3-cocycles of height 1.