Combinatorial cohomology of the space of long knots

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.