Equilevel algebras

An equilevel algebra is a subalgebra of the space of smooth functions $f: M \to {\mathbb R}$ distinguished in this space by finitely many linear conditions of the type $f(x_i) = f(\tilde x_i)$, $x_i \neq \tilde x_i \in M$, or approximated by such subalgebras. For $M = S^1$ or ${\mathbb R}^1$, the regular points of the variety of all equilevel subalgebras of codimension $k$ in $C^\infty(M, {\mathbb R})$ are known in knot theory as $k$-chord diagrams. The whole of this variety completes the space of chord diagrams in the same way as the spaces of ideals of given codimensions in complete the configuration spaces. We describe cell structures of the varieties of all equilevel algebras up to the codimension three in $C^\infty(S^1, {\mathbb R})$ and compute their homology rings and characteristic classes of canonical vector bundles. We also give lower estimates of these rings for arbitrary $k$.