Cohomology groups for projection tilings of codimension 2

The gap-labelling group, which provides the set of possible values of the integrated density of states on gaps in the spectrum of a Hamiltonian describing particles in a tiling, is frequently related to the cohomology of the tiling. We present explicit results for the cohomology of many well-known tilings obtained from the cut and projection method with codimension 2, including the (generalized) Penrose tilings, the Tübingen triangle tiling, the Ammann–Beenker tiling, and the Socolar tiling.