Post date: Jun 29, 2011 6:25:2 AM
A generic closed plane curve is a smooth immersion f:S1➝R2 such that its singularities are only transversal double points. In particular, it doesn′t have triple points and self-tangent points. Up to isotopy, there is only one curve with no double point, two plane curves with one double point, five plane curves with two double points etc.
Theory of finite type invariants of closed plane curves might be more reach and complicated than that of knots. The reason to say so is that there exist non-trivial 1st order Vasiliev invariants of plane curves, but there is none for knots.
Being a 3rd year student at Moscow University, I did a project on plane curves. I created a computer programme to calculate the number of topological types of plane curves with up to 10 double points. As a result, we (me and my supervisor Prof S. Gusein-Zade) published a paper in Russian Mathematical Surveys. Now I'm supervising an undergraduate research project on plane curves. We do some programming, too, on stuff like drawing curves and computations of first order invariants.
TEXTBOOK
[shop] Vladimir Arnold Topological Types of Plane Curves and Caustics (University Lecture Series)
REFERENCES
[ps.gz] S. Chmutov, S. Duzhin Explicit formulas for Arnold's generic curve invariants "Arnold-Gelfand Mathematical Seminars: Geometry and Singularity Theory." Birkhauser, 1997, pp. 123--138
DOWNLOADS
[pdf] S. Gusein-Zade, F. Duzhin On the number of topological types of plane curves, Russian Mathematical Surveys 53, pp. 626-627
LINKS
[url] NTU Curves - webpage of our undergraduate research project on computation of first order curves invariants.
[url] Sergei Duzhin's homepage