Functional Analysis and Its Applications

, Volume 40, Issue 1, pp 1–10

# J-invariants of plane curves and framed chord diagrams

• S. K. Lando
Article

## 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

## References

1. 1.
V. I. Arnold, Topological Invariants of Plane Curves and Caustics, Univ. Lecture Ser., Vol. 5, Amer. Math. Soc., Providence, RI, 1994.Google Scholar
2. 2.
V. I. Arnold, “Plane curves, their invariants, perestroikas and classifications,” In: Singularities and Bifurcations, Adv. Soviet Math., Vol. 21, Amer. Math. Soc.,Providence, RI, 1994, pp. 33–91.Google Scholar
3. 3.
V. I. Arnold, “Symplectization, complexification and mathematical trinities,” In: Fields Inst. Commun., Vol. 24, Amer. Math. Soc.,Providence, RI, 1999, pp. 23–37.Google Scholar
4. 4.
D. Bar-Natan, “On the Vassiliev knot invariants,” Topology, 34, 423–472 (1995).Google Scholar
5. 5.
I. Beck, “Cycle decompositions by transpositions,” J. Combin. Theory Ser. A, 23, 198–207 (1977).Google Scholar
6. 6.
R. Fenn and P. Taylor, “Introducing doodles,” In: Topology of Low-Dimensional Manifolds, R. Fenn ed., Lecture Notes in Math., Vol. 722, Springer-Verlag, Heidelberg, 1977, pp. 37–43.Google Scholar
7. 7.
V. Goryunov, “Vassiliev type invariants in Arnold’s J +-theory of plane curves without direct self-tangencies,” Topology, 37, 603–620 (1998).Google Scholar
8. 8.
J. W. Hill, “Vassiliev-type invariants of plane fronts without dangerous self-tangencies,” C. R. Acad. Sci. Paris Sér. Math., 324, No. 5, 537–542 (1997).Google Scholar
9. 9.
M. Kazaryan, “First order invariants of strangeness type for plane curves,” Trudy Mat. Inst. Steklov, 221, 213–224 (1998); English transl.: Proc. Steklov Inst. Math, No. 2 (221), 202–213 (1998).Google Scholar
10. 10.
R. Kirby, “A calculus of framed links in S 3,” Invent. Math., 45, 35–56 (1978).Google Scholar
11. 11.
B. Khesin and A. Rosly, “Symplectic geometry on moduli spaces of holomorphic bundles over complex surfaces,” In: Fields Inst. Commun., Vol. 24, Amer. Math. Soc.,Providence, RI, 1999, pp. 311–323.Google Scholar
12. 12.
S. Lando, “On a Hopf algebra in graph theory,” J. Combin. Theory. Ser. B, 80, 104–121 (2000).Google Scholar
13. 13.
A. Losev and Yu. Manin, “New moduli spaces of flat connections,” Michigan Math. J., 48, 443–472 (2000).Google Scholar
14. 14.
B. Mellor, “A few weight systems arising from intersection graphs,” Michigan Math. J., 51, No. 3, 509–536 (2003).Google Scholar
15. 15.
G. Moran, “Chords in a circle and a linear algebra over GF(2),” J. Combin. Theory Ser. A, 37, 239–247 (1984).Google Scholar
16. 16.
V. V. Prasolov and A. B. Sossinskii, Knots, Links, Braids and 3-Manifolds. An Introduction to the New Invariants in Low-Dimensional Topology, Transl. Math. Monographs, Vol. 154, Amer. Math. Soc., Providence, RI, 1997.Google Scholar
17. 17.
E. Soboleva, “Vassiliev knot invariants coming from Lie algebras and 4-invariants,” J. Knot Theory Ramifications, 10, 161–169 (2001).Google Scholar
18. 18.
V. A. Vassiliev, “Invariants of ornaments,” In: Singularities and Bifurcations, Adv. Soviet Math., Vol. 21, Amer. Math. Soc.,Providence, RI, 1994, pp. 225–262.Google Scholar
19. 19.
V. A. Vassiliev, Topology of Complements to Discriminants [in Russian], Fazis, Moscow, 1997.Google Scholar
20. 20.
V. A. Vassiliev, “On finite order invariants of triple point free plane curves,” In: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, Vol. 194, 1999, pp. 275–300.Google Scholar