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.
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