Polygon recutting as a cluster integrable system

TL;DR

This paper demonstrates that polygon recutting operations form a completely integrable system linked to cluster transformations, revealing deep algebraic structures and refactorization interpretations.

Contribution

It establishes that polygon recutting actions are cluster transformations and proves their complete integrability using quaternionic polynomial refactorization.

Findings

Recutting operations generate an affine symmetric group action.

The recutting action is shown to be a cluster transformation.

The integrability is proven via quaternionic polynomial refactorization.

Abstract

Recutting is an operation on planar polygons defined by cutting a polygon along a diagonal to remove a triangle, and then reattaching the triangle along the same diagonal but with opposite orientation. Recuttings along different diagonals generate an action of the affine symmetric group on the space of polygons. We show that this action is given by cluster transformations and is completely integrable. The integrability proof is based on interpretation of recutting as refactorization of quaternionic polynomials.