Equilevel algebras

Abstract

Singular knots are smooth maps $S^1 \to {\mathbb R}^3$ that have self-intersections or points at which the derivative vanishes. We describe an algebraic classification of their singularities, and study the topological properties of the corresponding stratification of the discriminant subset $\Sigma \subset C^\infty (S^1, {\mathbb R}^3)$ consisting of all singular knots. This classification is defined by a system of special subalgebras of the function space $C^\infty(S^1, {\mathbb R})$. These are either defined by the chord diagrams, that is, by finite sets of conditions of the form $f(x_i) = f(\tilde x_i)$, $\{x_i, \tilde x_i\} \subset S^1$, or are the limit positions of such subalgebras in the space of all subspaces of a fixed codimension in $C^\infty(S^1, {\mathbb R})$. These limits arise at various collisions of the points $x_i, \tilde x_i$ that define the chord diagrams. For each natural number $k$, the set of such codimension-$k$ subalgebras is a $2k$-dimensional compact semialgebraic variety with a canonical $k$-dimensional vector bundle on it. We describe the natural stratification of these varieties for $k \leq 3$ and compute their cohomology rings and characteristic classes of canonical vector bundles. We also find many cohomology classes of sets of these subalgebras of arbitrary codimensions, and prove geometric corollaries concerning the topology of corresponding discriminant strata, in particular on their intersections with the finite-dimensional approximating subspaces of the knot space.