Relaxed Cech Cohomology, Emeralds over Topological Spaces and the Kontsevich Integral

TL;DR

This paper develops a new cohomology framework called relaxed Cech cohomology, introduces the concept of emeralds over topological spaces, and connects these ideas to the Kontsevich integral in knot theory.

Contribution

It introduces the concept of emeralds over spaces and a relaxed Cech cohomology, providing a novel formalism linking topology, sheaves, and knot invariants.

Findings

Defined emeralds as families of decorated spaces and sheaves.

Established a relaxed Cech cohomology with intersection up to equivalence.

Connected the formalism to the Kontsevich integral in knot theory.

Abstract

We introduce families of decorations of a same topological space, as well as a family of sheaves over such decorated spaces. Making those families a directed system leads to the concept of emerald over a space. For the configuration space X_N of N points in the plane, connecting points of the plane with chords is a decoration and the sheaf of log differentials over such spaces forms an emerald. We introduce a relaxed form of Cech cohomology whereby intersections are defined up to equivalence. Two disjoint open sets of X_N whose respective points are connected by a chord is one instance of intersection up to equivalence. One paradigm example of such a formalism is provided by the Kontsevich integral.