Parallelogram morphisms and circular codes

Abstract

In 2014, it was conjectured that any polyomino can be factorized uniquely as a product of prime polyominoes [7]. In this paper, we present simple tools from words combinatorics and graph topology that seem very useful in solving the conjecture. The main one is called parallelogram network, which is a particular subgraph of G(ℤ2) induced by a parallelogram morphism, i.e. a morphism describing the contour of a polyomino tiling the plane as a parallelogram would. In particular, we show that parallelogram networks are homeomorphic to G(ℤ2). This leads us to show that the image of the letters of parallelogram morphisms is a circular code provided each element is primitive, therefore solving positively a 2013 conjecture [8].