Vassiliev invariants classify plane curves and doodles

Abstract

An ornament is a system of oriented closed curves in a plane or some other 2-surface no three of which intersect at one point. Similarly, a doodle is a collection of oriented closed curves without triple points or degenerations. Homotopy invariants of ornaments and doodles are natural analogues of homotopy and isotopy invariants of links, respectively. The Vassiliev theory of finite-order invariants of ornaments and the constructions of certain series of such invariants can be applied to doodles. It is proved that these finite-order invariants classify doodles. Similar finite-order invariants of connected oriented closed curves classify doodles up to an isotopy of the ambient plane.