Ribbon Tile Invariants from the Signed Area

Ribbon tiles are polyominoes consisting of n squares laid out in a path, each step of which goes north or east. Tile invariants were first introduced by the second author (2000, Trans. Amer. Math. Soc.352, 5525–5561), where a full basis of invariants of ribbon tiles was conjectured. Here we present a complete proof of the conjecture, which works by associating ribbon tiles with certain polygons in the complex plane, and deriving invariants from the signed area of these polygons.