Combinatorial cohomology of the space of long knots

TL;DR

This paper introduces a combinatorial cohomology framework for the space of long knots, simplifying the identification and evaluation of cohomology classes through linear algebra and explicit algebraic intersections.

Contribution

It develops a combinatorial graded cochain complex that enables easy computation of cohomology classes in the knot space, extending arrow diagram techniques.

Findings

Defines a combinatorial graded cochain complex for knot cohomology

Constructs explicit algebraic intersections with simple strata in knot space

Provides canonical co-orientations for these strata

Abstract

The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We achieve this goal by defining a combinatorial graded cochain complex, such that the elements of an explicit submodule in the cohomology define algebraic intersections with some "geometrically simple" strata in the space of knots. Such strata are endowed with explicit co-orientations, that are canonical in some sense. The combinatorial tools involved are natural generalisations (degeneracies) of usual methods using arrow diagrams.