Abstract
Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. Periodic isotopy classifications are obtained for various families of embedded nets with small quotient graphs. The 25 periodic isotopy classes of depth-1 embedded nets with a single-vertex quotient graph are enumerated. Additionally, a classification is given of embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, as well as demonstrations of their maximal symmetry periodic isotopes. The methodology of linear graph knots on the flat 3-torus [0,1)3 is introduced. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.
Highlights
Entangled and interpenetrating coordination polymers have been investigated intensively by chemists in recent decades
The net N is an embedded net for a topology G, it is translationally periodic with respect to each basis vector of some vector space basis for the ambient space, the nodes are distinct points, and the bonds of N are noncolliding straight-line segments between nodes
Subclasses of linear d-periodic nets N in Rd are defined in terms of the diversity of their connected components and we indicate the connections between these class divisions and those used for entangled coordination polymers
Summary
Entangled and interpenetrating coordination polymers have been investigated intensively by chemists in recent decades Their classification and analysis in terms of symmetry, geometry and topological connectivity is an ongoing research direction (Batten & Robson, 1998; Carlucci et al, 2003, 2014; Blatov et al, 2004; Alexandrov et al, 2011). Such multicomponent embedded nets are related to the interpenetrated structures with translationally equivalent components which are abundant in coordination polymers.
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