Construction of weaving and polycatenane motifs from periodic tilings of the plane

From microscopic molecular weaving to macroscopic technical fabrics, two-directional periodic entanglement is common across scales and exhibits rich mathematical structure. Doubly periodic (DP) weaves and polycatenanes are intricate entangled structures. They are embedded in the Euclidean thickened plane and remain invariant under translations along two independent directions. Their topological properties are fully encoded within a quotient space under a periodic lattice, which we refer to as a motif. On the diagrammatic level, a motif diagram is a specific type of link diagram in the torus. It consists of essential simple closed curves in the case of DP weaves, or null-homotopic curves in the case of DP polycatenanes. In this paper, we introduce a combinatorial methodology to construct these motifs from planar DP tilings, using the concept of polygonal link transformations. We also present an approach to predict the type of motif that can be constructed from a given DP tiling and a chosen polygonal link method. This approach has potential applications in various disciplines, such as materials science and chemistry.