This paper introduces the key concepts and problems of the new research area of Periodic Geometry and Topology for applications in Materials Science. Periodic structures such as solid crystalline materials or textiles were previously studied as isolated structures without taking into account the continuity of their configuration spaces. The key new problem in Periodic Geometry is an isometry classification of periodic point sets. A required complete invariant should continuously change under point perturbations, because atoms always vibrate in real crystals. The main objects of Periodic Topology are embeddings of curves in a thickened plane that are invariant under lattice translations. Such periodic knots were classified in the past up to continuous deformations (isotopies) that keep a fixed lattice structure, hence are realized in a fixed thickened torus. The more practical equivalence is a periodic isotopy in a thickened plane without fixing a lattice basis. The paper states the first results in the new area and proposes further problems and directions.
翻译:本文介绍了用于材料科学应用的定期几何和地形学新研究领域的关键概念和问题; 固体晶体材料或纺织品等定期结构以前作为孤立结构研究,没有考虑到其配置空间的连续性; 周期几何中的关键新问题是定期点数组的异度分类; 所需的完整变异性应在点扰动下不断改变, 因为原子总是在真实晶体中振动; 周期地形学的主要对象将曲线嵌入一个厚厚的平面中,而这种平面在变异翻译中是不可变的。 这种周期结结结在过去被分类为连续的变形( 软结), 保持固定的阵形结构, 从而在固定的厚度中实现。 更实际的等同性是在厚的平面中进行定期的不固定的异性, 而没有固定的粘结基。 论文描述了新区域的第一个结果, 并提出进一步的问题和方向 。