Winding Numbers and Topology in Aperiodic Tilings

Abstract Structural and spectral properties of periodic tilings are well understood and related by the Bloch theorem. For quasiperiodic tilings generated by the Cut and Project (C&P) procedure, we show how to relate structure and spectrum by means of topological numbers originated in windings of a certain phase. For general non-C&P tilings, we illustrate how to establish structural features based on cohomology groups, and present an extension of the Bloch theorem based on these groups.