Arnold invariants

In mathematics, particularly in topology and knot theory, Arnold invariants are invariants introduced by Vladimir Arnold in 1994[1] for studying the topology and geometry of plane curves.

—provide ways to classify and understand how curves can be deformed while preserving certain properties.

[2] The fundamental context for Arnold invariants comes from the Whitney–Graustein theorem, which states that any two immersed loops (smooth curves in the plane) with the same rotation number can be deformed into each other through a series of continuous transformations.

[3] These transformations can be broken down into three elementary types: direct self-tangency moves (where two portions of the curve become tangent with aligned directions, either creating or eliminating two self-intersection points), inverse self-tangency moves (similar to direct moves, but the tangent directions are opposite), and triple point moves (where three portions of the curve intersect at a single point).

invariants keep track of how curves change under these transformations and deformations.

invariant increases by 2 when a direct self-tangency move creates new self-intersection points (and decreases by 2 when such points are eliminated), while

decreases by 2 when an inverse self-tangency move creates new intersections (and increases by 2 when they are eliminated).

Neither invariant changes under triple point moves.

A fundamental relationship between these invariants is that their difference equals the total number of self-intersection points in the curve.

That is, Mathematicians Oleg Viro and Eugene Gutkin discovered an explicit formula for calculating

is the winding number around a point in region

[4] In 2002, Spanish mathematicians Mendes de Jesus and Romero Fuster introduced the concepts of bridges and channels for plane curves to facilitate the calculation of Arnold invariants.

This decomposition technique is particularly powerful for analyzing curves with double points.

An important theorem regarding this decomposition states that a curve with

with bridges having no double points, or a decomposition into exactly

(isotopic to the circle) with bridges having double points.

[7] This result proved a conjecture originally proposed by Arnold regarding the formulas for families of tree-like curves.

The bridge and channel technique provides a systematic method for computing Arnold invariants for plane curves in terms of simpler curves with at most one double point.