Topology of two-connected graphs and homology of spaces of knots

Abstract

We propose a new method of computing cohomology groups of spaces of knots in ${\mathbb{R}}^n$, $n\ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we define the higher indices, which invariants of knots in ${\mathbb{R}}^3$ define at arbitrary singular knots. More generally, for any finite-order cohomology class of the space of knots we define its principal symbol, which lies in a cohomology group of a certain finite-dimensional configuration space and characterizes our class modulo the classes of smaller filtration.