  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.
