V.Vassiliev
Abstract of the talk at Arnold's seminar, 
November 3, 1998

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{\bf On finite order invariants of triple points free
plane curves}
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The intensive study of invariants of generic immersions $S^1 \to {\bf R}^2$
was started by Arnold and continued by Viro, Polyak, Aicardi, Shumakovich, 
Tabachnikov, Gusein-Zade, Merkov, Kazaryan, and many others.

The most interesting invariant of such objects, the {\em strangeness}, 
is in fact an invariant of triple points free immersions 
$S^1 \to {\bf R}^2$ (with allowed self-tangencies).

We describe the regular techniques of calculating all finite-order
invariants of triple points free smooth plane curves
$S^1 \to {\bf R}^2$ (maybe with singular points: a problem previously
considered also by A.~B.~Merkov, who obtained some basic results).
These techniques are a direct analog of these for knot invariants and 
are based on the calculus of {\em triangular diagrams} and {\em connected
hypergraphs} in the same way as the calculation of knot invariants
is based on the study of chord diagrams and connected graphs.

E.g., the simplest such invariant is of order 4 and corresponds to the
triangular diagram
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\linethickness{0.4pt}
\begin{picture}(3.33,4.34)
\put(0.67,1.67){\circle{5.33}}
\put(0.67,-1.00){\line(-3,5){2.33}}
\put(-1.66,3.00){\line(1,0){4.33}}
\put(2.67,3.00){\line(-1,-2){2.00}}
\put(0.67,4.34){\line(-1,-2){2.00}}
\put(-1.33,0.34){\line(1,0){4.00}}
\put(2.67,0.34){\line(-1,2){2.00}}
\end{picture}
in the same way as the simplest knot invariant (of order 2)
corresponds to the 2-chord diagram $\bigoplus$.
Also, following V.~I.~Arnold, we consider invariants of
{\em immersed} triple points free curves and describe similar techniques
also for this problem, and, more generally, for the calculation
of homology groups of the space of immersed plane curves without
points of multiplicity $\ge k$ for any $k \ge 3.$

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