Equilevel algebras

Singular knots are smooth maps [Formula: see text] 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 [Formula: see text] consisting of all singular knots. This classification is defined by a system of special subalgebras of the function space [Formula: see text], which we call the equilevel algebras. These are either defined by the chord diagrams, that is, by finite sets of conditions of the form [Formula: see text], [Formula: see text], or are the limit positions of such subalgebras within the space of subspaces of finite codimension in [Formula: see text]. These limits arise at various collisions of the points [Formula: see text] that define the chord diagrams. For each natural number [Formula: see text], the set of such codimension-[Formula: see text] subalgebras is a [Formula: see text]-dimensional compact stratified variety with a canonical [Formula: see text]-dimensional vector bundle on it. We describe the natural stratification of these varieties for [Formula: see text] 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.