Isotopy classification of three-dimensional embedded nets

Abstract

Nets are important mathematical tools that have been used as models in the analysis and design of crystal structures since the early days of crystallography (Wells, 1977). The notion of a net is extremely versatile and lends itself to major applications in crystal engineering and crystal structure prediction, including metal-organic frameworks (MOFs) and zeolites. Apart from this wide range of applicability, nets and their embeddings are also fascinating mathematical objects in their own right and have significant applications in mathematics itself, including in polyhedral geometry, tiling theory, minimal surfaces and discrete symmetry, to name a few.The underlying topological structure of a net is provided by an abstract graph G, with finite or (usually) countably infinite vertex set and edge set, and in the infinite case equipped with an abstract periodicity structure determined by a free abelian subgroup of the combinatorial automorphism group of G. The infinite graphs come to life through their 3-periodic embeddings as periodic bond-node structures N in an ambient geometric space, which in most applications is ordinary Euclidean 3-space R 3 . These embedded nets N can have several different characteristics. In the embedding, the vertices of G are represented by (usually distinct) points, called the vertices (or nodes) of N , and the edges of G are represented by what are called edges (or bonds) of N , which usually are straight line segments (sometimes with a specified length), possibly allowed to collide, or sometimes are non-self-intersecting paths of various sorts. The entire embedded net N is required to be invariant under a group of translations T of the ambient space generated by the translations in the directions of the basis vectors from a vector-space basis, called the periodicity basis of N . This translation group T provides a geometric representation of the abstract periodicity structure on the underlying structure graph. In a proper linear 3-periodic net N in R 3 , the edges of N are noncolliding straight line segments joining vertices of N . In applications, the vertices of an embedded net N may represent spatial positions of atoms or of clusters of atoms, such as single metal ions or secondary building units, and the edges may represent chemical bonds or ligands. A net can represent many compounds.My own interest in nets relates to discrete geometry and was inspired by the pioneering contributions of Michael O'Keeffe and his collaborators (see, for example, Delgado-Friedrichs et al., 2005). These included Omar Yaghi, who was a co-founder of reticular chemistry with O'Keeffe (for which they were jointly awarded the Gregori Aminoff Prize in Crystallography by The Royal Swedish Academy of Sciences in 2019), and Davide Proserpio, to whom I am grateful for making me aware of the extensive works on crystal nets and related geometric structures in the chemistry literature (and for kindly providing the two figures). Several databases have been created to keep track of the vast amount of information on ...