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 R, n ≥ 3, generalizing the Polyak—Viro formulas [9] for invariants (i.e., 0-dimensional cohomology classes) of knots in R. 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 ↪→ R 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 ↪→ R. As a corollary, we prove the nontriviality of all these cohomology classes in spaces of knots in R. 2000 Math. Subj. Class. Primary 57M25, 55R80; Secondary 57Q45, 55T99, 54F05.