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Article

Combinatorial formulas for cohomology of knot spaces

Moscow Mathematical Journal. 2001. Vol. 1. No. 1. P. 91-123.
We develop homological techniques for finding explicit combinatorial expressions of finite-type cohomology classes of spaces of knots in Rn,n≥3, generalizing Polyak--Viro formulas for invariants (i.e. 0-dimensional cohomology classes) of knots in R3. As the first applications we give such formulas for the (reduced mod 2) {\em generalized Teiblum--Turchin cocycle} of order 3 (which is the simplest cohomology class of {\em long knots} R1↪Rn 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} S1↪Rn. As a corollary, we prove the nontriviality of all these cohomology classes in spaces of knots in R3.