Combinatorial formulas for cohomology of knot spaces

We develop homological techniques for finding explicit combinatorial formulas for finite-type cohomology classes of spaces of knots in $\mathbb R^n$, $n\ge 3$, generalizing the Polyak–Viro formulas [PV] for invariants (i.e., 0-dimensional cohomology classes) of knots in $\mathbb{R}^3$. 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 $\mathbb R^1\hookrightarrow\mathbb R^n$ not reducible to knot invariants or their natural stabilizations), and for all integral cohomology classes of orders 1 and 2 of spaces of compact knots $S^1\hookrightarrow\mathbb R^n$. As a corollary, we prove the nontriviality of all these cohomology classes in spaces of knots in $\mathbb R^3$.