Arnold Invariants

In mathematics, particularly in topology and knot theory, Arnold invariants are invariants introduced by Vladimir Arnold in 1994Arnold, V. I. (1994). ''Topological Invariants of Plane Curves and Caustics''. University Lecture Series, Vol. 5, American Mathematical Society. for studying the topology and geometry of plane curves. The three main invariants—$J^+$, $J^-$, and $St$—provide ways to classify and understand how curves can be deformed while preserving certain properties.Mai, Alexander (2022). "Introduction to Arnold's J+-Invariant". arXiv:2210.00871.

Background

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.Whitney, H. (1937). "On regular closed curves in the plane". Compositio Mathematica, 4, 276-284. 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).Moraes, Simone (2018). "Invariants of Closed Plane Curves". Federal University of Bahia.

''J^±'' invariants

The $J^+$ and $J^-$ invariants keep track of how curves change under these transformations and deformations. The $J^+$ 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 $J^-$ 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,
:$J^+(c) - J^-(c) = \textc$.Professor Paul Seidel online lecture notes at https://ocw.mit.edu/courses/18-900-geometry-and-topology-in-the-plane-spring-2023/mit18_900s23_lec17.pdf

Mathematicians Oleg Viro and Eugene Gutkin discovered an explicit formula for calculating $J^-$:
:$J^-(c) = 1 - \sum_R \text(c,R)^2 + \sum_q \text(c,q)^2$
where $R$ ranges over the regions into which $c$ divides the plane, $\text(c,R)$ is the winding number around a point in region $R$, and $\text(c,q)$ is the mean winding number at each self-intersection point $q$. For example, a curve with $k$ curls in standard form has $J^+ = -2k$ and $J^- = -3k$, while a simple circle has $J^+ = J^- = 0$.

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.Mendes de Jesus, C.; Romero Fuster, M. C. (2002). "Bridges, channels and Arnold's invariants for generic plane curves". '' Topology and its Applications'', 125, 505-524. 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 $n$ double points is a tree-like curve if and only if it admits a decomposition into exactly n curves of types $K_0$ and $K_2$ with bridges having no double points, or a decomposition into exactly $n+1$ curves of type $K_1$ ( isotopic to the circle) with bridges having double points.Aicardi, F. (1994). "Tree-like Curves". In: Singularities and Bifurcations. Advances in Soviet Mathematics, 21, AMS, Providence, 1-36. 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.

See also

* Plane curve
* Knot invariant
* Whitney–Graustein theorem
* Differential topology

References

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Further reading

*Santa Rosa, Lílian Neves (2010). ''Arnold Invariants of Plane Curves''. Master's Thesis, Federal University of Viçosa.
*Mendes de Jesus, C. ''Topological Invariants of Generic Maps from Oriented Compact Surfaces to the Plane''. Doctoral Thesis, PUC-RIO, 2001.
*Moraes, Simone M.; Sánchez, Catarina M. J. "Invariants of Closed Plane Curves". ''Proceeding Series of the Brazilian Society of Applied and Computational Mathematics'', Vol. 3, N. 1, 2015.
*Lagemann, Anna Marie; von der Mosel, Heiko (Thesis advisor); Hryniewicz, Umberto (Thesis advisor); Reiter, Philipp (Thesis advisor).
Variational approach to study Arnold’s invariants of immersed planar curves
' RWTH Aachen University (2023) PhD Thesis

Original source

*Vladimir Arnold:
Plane curves, Their Invariants, Perestroikas and Classifications
. 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.

Geometry
Differential topology
Knot theory
Knot invariants
Differential geometry
Symplectic geometry