Quandle cohomology theory was developed [5] to define invariants of classical knots and knotted surfaces in state-sum form, called quandle cocycle (knot) invariants. The quandle cohomology theory is a modification of rack cohomology theory which was defined in [11]. The cocycle knot invariants are analogous in their definitions to the Dijkgraaf-Witten invariants [8] of triangulated 3-manifolds ...

Given a biquandle [Formula: see text], function text] with certain compatibility and pair of non commutative cocyles values in necessarily group we give an invariant for singular knots/links. also define universal functions governing all 2-cocycles exhibit examples computations. When the target is abelian, notion abelian cocycle given “state sum” defined Computations generalizing linking number...

Let $\Gamma$ be an irreducible lattice in a product of two locally compact groups and assume that is densely embedded profinite group $K$. We give necessary conditions which imply the left translation action $\Gamma \curvearrowright K$ “virtually” cocycle superrigid: any ${w\colon \Gamma\times K\rightarrow\Delta}$ with values countable $\Delta$ cohomologous to factors through map $\Gamma\times ...

We consider continuous SL(2,R)-cocycles over a minimal homeomorphism of a compact set K of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform subexponential growth.

Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.