Functional Analysis and Its Applications

, Volume 40, Issue 1, pp 1–10 | Cite as

J-invariants of plane curves and framed chord diagrams

  • S. K. Lando


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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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • S. K. Lando
    • 1
    • 2
  1. 1.Poncelet Laboratory of the Independent University of MoscowRussia
  2. 2.Institute for Systems ResearchRussian Academy of SciencesRussia

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