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

Abstract: Doubly periodic (DP) weaves and polycatenanes are complex entangled structures embedded in the Euclidean thickened plane, invariant under translations in 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 is a specific type of link diagram on the torus, consisting of essential closed curves for DP weaves or null-homotopic curves for 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.