Introduction to Periodic Geometry and Topology

Abstract

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.