On Finite Order Invariants of Triple Points Free Plane Curves

TL;DR

This paper develops techniques for calculating finite order invariants of triple points free plane curves, drawing analogies with knot invariants, and extends these methods to immersed curves and homology groups of curve spaces.

Contribution

It introduces regular techniques based on triangular diagrams and hypergraphs for computing invariants of triple points free plane curves, extending to immersed curves and homology calculations.

Findings

A degree 4 invariant corresponding to two triangles in a diagram.

Extension of techniques to immersed curves without triple points.

Description of homology groups of spaces of immersed plane curves.

Abstract

We describe some regular techniques of calculating finite degree invariants of triple points free smooth plane curves $S^1 \to R^2$. They are a direct analog of similar techniques 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 degree 4 and corresponds to the diagram consisting of two triangles with alternating vertices in a circle in the same way as the simplest knot invariant (of degree 2) corresponds to the 2-chord diagram $\bigoplus$. Also, following V.I.Arnold and other authors 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 $k \ge 3.$