On combinatorial formulas for cohomology of spaces of knots

Victor A. Vassiliev

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arXiv · 1407.7235

On combinatorial formulas for cohomology of spaces of knots

Victor A. Vassiliev

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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 $R^3.$

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Victor A. Vassiliev. 2014-07-27. On combinatorial formulas for cohomology of spaces of knots. https://arxiv.org/abs/1407.7235

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