V.A.Vassiliev
On combinatorial formulas for cohomology of spaces of knots
25-Sep-2000

Abstract

We develop homological techniques for finding explicit combinatorial
formulas for finite-type cohomology classes of spaces of knots in 
$\R^n, n\ge 3,$ generalizing Polyak--Viro formulas for invariants (i.e.
0-dimensional cohomology classes) of knots in $\R^3$.

As the first applications we give such formulas for the (reduced mod 2)
generalized Teiblum--Turchin cocycle of order 3 (which is the simplest
cohomology class of long knots $\R^1 \hookrightarrow \R^n$ not reducible to
knot invariants or their natural stabilizations), and for all integral
cohomology classes of orders 1 and 2 of spaces of {\em compact knots} $S^1
\hookrightarrow \R^n$. As a corollary, we prove nontriviality of all these
cohomology classes in spaces of knots in $\R^3.$