arXiv research

The paper studies signature cocycles on mapping class groups and symplectic groups.

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…

Extends a formula for the homomorphism defect of a signature map to coloured braids.

Formula for signature of handlebody bundles, interpreting cohomology.

Taking the signature of the closure of a braid defines a map from the braid group to the integers. In 2005, Gambaudo and Ghys expressed the homomorphism defect of this map in terms of the Meyer cocycle and the Burau representation. In the present paper, we simultaneously extend this result in two directions, considerin…

Unified framework for studying Torelli group and congruence subgroup maps.

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 …

In this paper we define a $\mathbf{QP}^1$-valued class function on the mapping class group $\mathcal{M}_{g,2}$ of a surface $Σ_{g,2}$ of genus $g$ with two boundary components. Let $E$ be a $Σ_{g,2}$ bundle over a pair of pants $P$. Gluing to $E$ the product of an annulus and $P$ along the boundaries of each fiber, we …

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.

Study projective derivative cocycles for circle diffeomorphisms.

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…

New graph-based invariants from quandle cocycles.

Virtual index cocycles reformulate 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…

Study on group cocycles for volume-preserving diffeomorphisms.

The paper extends Euler class theory to measurable cocycles.

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…

New rack and multiple group rack cohomology for surfaces in 3-sphere.

Researchers compute a residue cocycle for Dirac-type operators using modified Getzler calculus.

New algebraic rules for 5D shapes based on 3D cocycles.

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…

Quantum cocycle invariants derived from Yang-Baxter cohomology.

New shifting chain map enhances quandle invariants for links.

Develops Patterson-Sullivan theory for coarse cocycles.

We explore a knot invariant derived from colorings of corresponding $1$-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle $2$-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles w…

Enhances psyquandle counting invariants using cocycles.

Paper extends multiplicative constants to measurable cocycles theory.

Study of Penner's cocycle on fatgraph complex.

Study of quandle coloring quivers with dihedral quandles.

New invariant for spin 3-manifolds using super 3-cocycles.

Paper studies quandle shadow cocycle invariants and Vassiliev invariants.

Paper studies knotoid chirality using shadow quandle colorings and invariants.

Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial $2$-cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic condit…

Study parametrized Kähler class for cocycles on Hermitian symmetric spaces.

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…

The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram. Such an interpretation is applied to evaluating the invariants. Algebraic inter…

This paper defines maximal measurable cocycles for surface groups into Hermitian Lie groups and studies their algebraic hulls.

We describe all the situations in which the Kontsevich-Zorich cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, this only occurs when the cocycle satisfies additional geometric constraints. We also describe the real Lie groups which can appear in the monodromy of the Kontsevich…

Classifies central extensions for area-preserving diffeomorphisms and shows they are fuzzy sphere limits.

The paper constructs non-trivial cocycles for long embeddings with more than one loop.

New volume invariant for cocycles of hyperbolic lattices, proving rigidity results.

Study lengths of 3-cocycles for specific quandles, finding knot properties.

Graph morphism maps Poisson cocycles to symmetries, revealing factorization through Jacobi identity.

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 $Φ^{\pm}_φ(K)$ is trivial for all knots $K$.

Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow worl…