On combinatorial formulas for cohomology of spaces of knots

TL;DR

This paper develops combinatorial formulas for finite-type cohomology classes of knot spaces in higher dimensions, extending known invariants and proving their nontriviality in R^3.

Contribution

It introduces explicit combinatorial expressions for higher-order cohomology classes of knot spaces, generalizing Polyak--Viro formulas to spaces of knots in R^n.

Findings

Formulas for the generalized Teiblum--Turchin cocycle of order 3

Formulas for all integral cohomology classes of orders 1 and 2 for compact knots

Proof of nontriviality of these classes in R^3

Abstract

We develop homological techniques for finding explicit combinatorial expressions of 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) {\em generalized Teiblum--Turchin cocycle} of order 3 (which is the simplest cohomology class of {\em 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 the nontriviality of all these cohomology classes in spaces of knots in $R^3.$