Moduli-space structure of knots with intersections

Abstract

It is well known that knots are countable in ordinary knot theory. Recently, knots with intersections have raised a certain interest, and have been found to have physical applications. We point out that such knots—equivalence classes of loops in R3 under diffeomorphisms—are not countable; rather, they exhibit a moduli-space structure. We characterize these spaces of moduli and study their dimension. We derive a lower bound (which we conjecture being actually attained) on the dimension of the (nondegenerate components) moduli spaces, as a function of the valence of the intersection.