Functional characterization of Vasil'ev invariants

A family of subsets of a manifold $X$ is called an $r$-cover of $X$ if any $r$ points of $X$ are contained in a set in the family. Let $X$ and $Y$ be two smooth manifolds, $\operatorname{Emb}(X,Y)$ the family of smooth embeddings, $M$ an Abelian group, and $F\colon\operatorname{Emb}(X,Y)\to M$ a functional. We say that $F$ has degree not greater than $r$ if for each finite open $r$-cover $\{U_i\}_{i\in I}$ of $X$ there exist functionals $F_i\colon\operatorname{Emb}(U_i,Y)\to M$, $i\in I$, such that for each $a\in\operatorname{Emb}(X,Y)$ we have
$$ F(a)=\sum_{i\in I}F_i(a|_{U_i}). $$

The main result is as follows.
Theorem. {\it An isotopy invariant $F\colon\operatorname{Emb}(S^1,\mathbb R^3)\to M$ has finite degree if and only if $F$ is a Vasil'ev invariant. If $F$ is a Vasil'ev invariant of order $r$, then the degree of $F$ is equal to $2r$.}
Bibl. – 3 titles.