Rhombus Tilings: Decomposition and Space Structure

We study the spaces of rhombus tilings, i.e. the graphs whose vertices are tilings of a fixed zonotope, and two tilings are linked if one can pass from one to the other one by a local transformation, called flip. We first use a decomposition method to encode rhombus tilings and give a useful characterization for a sequence of bits to encode a tiling. In codimension 2, we use the previous coding to get a canonical representation of tilings, and an order structure on the space of tilings, which is shown to be a graded poset, from which connectivity is deduced.