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.
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