*J*-invariants of plane curves and framed chord diagrams

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## Abstract

Arnold defined *J*-invariants of general plane curves as functions on classes of such curves that jump in a prescribed way when passing through curves with self-tangency. The coalgebra of framed chord diagrams introduced here has been invented for the description of finite-order *J*-invariants; it generalizes the Hopf algebra of ordinary chord diagrams, which is used in the description of finite-order knot invariants. The framing of a chord in a diagram is determined by the type of self-tangency: direct self-tangency is labeled by 0, and inverse self-tangency is labeled by 1. The coalgebra of framed chord diagrams unifies the classes of *J*^{+}-and *J*^{−}-invariants, so far considered separately. The intersection graph of a framed chord diagram determines a homomorphism of this coalgebra into the Hopf algebra of framed graphs, which we also introduce. The combinatorial elements of the above description admit a natural complexification, which gives hints concerning the conjectural complexification of Vassiliev invariants.

## Key words

plane curve*J*-invariant invariant of finite order chord diagram framed chord diagram framed graph Hopf algebra of graphs

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