$J$-Invariants of Plane Curves and Framed Chord Diagrams

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.