Winding Numbers and Topology of Aperiodic Tilings

We show that diffraction features of $1D$ quasicrystals can be retrieved from a single topological quantity, the \v{C}ech cohomology group, $\check{H}^{1}\cong\mathbb{Z}^2$, which encodes all relevant combinatorial information of tilings. We present a constructive way to calculate $\check{H}^{1}$ for a large variety of aperiodic tilings. By means of two winding numbers, we compare the diffraction features contained in $\check{H}^{1}$ to the gap labeling theorem, another topological tool used to label spectral gaps in the integrated density of states. In the light of this topological description, we discuss similarities and differences between families of aperiodic tilings, and the resilience of topological features against perturbations.