Arnold invariants


In mathematics, particularly in topology and knot theory, Arnold invariants are invariants introduced by Vladimir Arnold in 1994 for studying the topology and geometry of plane curves. The three main invariants—,, and —provide ways to classify and understand how curves can be deformed while preserving certain properties.

Background

The fundamental context for Arnold invariants comes from the Whitney–Graustein theorem, which states that any two immersed loops with the same rotation number can be deformed into each other through a series of continuous transformations. These transformations can be broken down into three elementary types: direct self-tangency moves, inverse self-tangency moves, and triple point moves.

''J±'' invariants

The and invariants keep track of how curves change under these transformations and deformations. The invariant increases by 2 when a direct self-tangency move creates new self-intersection points, while decreases by 2 when an inverse self-tangency move creates new intersections. 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 :
where ranges over the regions into which divides the plane, is the winding number around a point in region, and is the mean winding number at each self-intersection point. For example, a curve with curls in standard form has and, while a simple circle has.

Bridges and channels

In 2002, mathematicians Catarina Mendes de Jesus and Maria Carmen Romero Fuster introduced the concepts of bridges and channels for plane curves to facilitate the calculation of Arnold invariants. A bridge consists of introducing a rectangle in the complement of the curve in the plane while respecting orientations, decomposing a given curve into two smaller curves with known invariants. The invariant of the original curve can then be obtained as a function of the invariants of these two component curves and the index of the bridge relative to the original curve. This decomposition technique is particularly powerful for analyzing curves with double points.
An important theorem regarding this decomposition states that a curve with double points is a tree-like curve if and only if it admits a decomposition into exactly n curves of types and with bridges having no double points, or a decomposition into exactly curves of type with bridges having double points. 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.

Original source

  • Vladimir Arnold: "". In Singularities and bifurcations, Vol. 21 of Adv. Soviet Math., Amer. Math. Soc., Providence, RI, 1994, pp. 33–91, with an appendix by F. Aicardi.