We generate a specific class of patterns by acting with a discrete symmetry group of the strip, hence a frieze group, on a collection of seeding shapes. If these shapes are actual discrete resonators carrying internal resonant modes and if the physics involved in the coupling of these modes is Galilean invariant, we demonstrate that all dynamical matrices responsible for the collective dynamics come from the regular representation of the stabilized frieze group $C^\ast$-algebra. As a result, the spectral bands of such dynamical matrices carry a complete set of topological invariants that can be resolved by the K-theory of this algebra, which is computed here together with an explicit basis. Based on these computations, we generate complete sets of topological models for all seven classes of frieze groups and demonstrate how to use them for topological spectral engineering of metamaterials.

中文翻译：

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带楣组的拓扑谱带

我们通过与条带的离散对称组（因此是带状组）作用于一组种子形状来生成特定类别的图案。如果这些形状是带有内部谐振模式的实际离散谐振器，并且如果这些模式耦合所涉及的物理是伽利略不变的，我们证明所有负责集体动力学的动力学矩阵都来自稳定楣组 $C^ 的正则表示\ast$-代数。因此，这种动态矩阵的谱带带有一组完整的拓扑不变量，可以通过该代数的 K 理论来解决，这里与显式基一起计算。基于这些计算，