非周期拼接的绕数与拓扑
Winding Numbers and Topology of Aperiodic Tilings
摘要：
我们发现，$1D$准晶的衍射特征可以从一个单一的拓扑量，Čech上同调群，$\check{H}^{1}\cong\mathbb{Z}^2$，它编码所有相关的组合信息的平铺检索。我们提出了一个建设性的方法来计算$\check{H}^{1}$为各种各样的非周期性平铺。通过两个缠绕数，我们比较了包含在$\check{H}^{1}$的衍射特性的间隙标记定理，另一个拓扑工具，用于标记光谱间隙的积分态密度。在这种拓扑描述的光，我们讨论了家庭之间的相似性和差异的非周期性平铺，和弹性的拓扑功能对扰动。
英文摘要：
We show that diffraction features of $1D$ quasicrystals can be retrieved from a single topological quantity, the Čech 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.
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