Research Article: Isotopy classes for 3-periodic net embeddings

Date Published: May 01, 2020

Publisher: International Union of Crystallography

Author(s): Stephen C. Power, Igor A. Baburin, Davide M. Proserpio.


Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type.

Partial Text

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 ▸). These investigations also draw on mathematical methodologies concerned with periodic graphs, group actions and classification (Delgado-Friedrichs, 2005 ▸; Koch et al., 2006 ▸; Schulte, 2014 ▸; Bonneau & O’Keeffe, 2015 ▸; Baburin, 2016 ▸). On the other hand, it seems that there have been few investigations to date on the dynamical aspects of entangled periodic structures with regard to deformations avoiding edge collisions, or with regard to excitation modes and flexibility in the presence of additional constraints (Guest et al., 2014 ▸). In what follows we take some first steps in this direction and along the way obtain some systematic classifications of basic families.

In any investigation with cross-disciplinary intentions, in our case between chemistry (reticular chemistry and coordination polymers) and mathematics (isotopy types and periodic frameworks), it is important to be clear of the meaning of terms. Accordingly we begin by defining all terminology from scratch.

We now define quotient graphs (QGs) and labelled quotient graphs (LQGs) associated with the periodic structure bases of a linear periodic net . Although QGs and LQGs are not sensitive to entanglement they nevertheless offer a means of subcategorizing linear periodic nets. See, for example, the discussions by Eon (2011 ▸, 2016 ▸), Klee (2004 ▸), Klein (2012 ▸), Thimm (2004 ▸) and Section 4.3 below.

We now define the adjacency depth of a linear 3-periodic net . This positive integer can serve as a useful taxonomic index and in Sections 9, 10 we determine, in the case of some small quotient graphs, the 3-periodic graphs which possess an embedding as a (proper) linear 3-periodic net with depth 1. These identifications also serve as a starting point for the determination of the periodic isotopy types of more general depth-1 embedded nets.

Let H be a multigraph, that is, a general finite graph, possibly with loops and with an arbitrary multiplicity of ‘parallel’ edges between any pair of vertices. Then a graph knot in is a faithful geometric representation of H where the vertices v are represented as distinct points p(v) in and each edge e with vertices v, w is represented by a smooth path , with endpoints p(v), p(w). Such paths are required to be free from self-intersections and disjoint from each other, except possibly for coinciding endpoints. Thus a graph knot K is formally a triple , and we may also refer to this triple as a spatial graph or as a proper placement of H in . It is natural also to denote a graph knot K simply as a pair (N, S), where N is the set of vertices, or nodes, p(v) in , and S is the set of nonintersecting paths . We remark that spatial graphs feature in the mathematical theory of intrinsically linked connected graphs (Conway & Gordon, 1983 ▸; Kohara & Suzuki, 1992 ▸).

Consider the following informal question: when can be deformed into by a continuous path with no edge crossings?

We now give some useful group-theoretic perspectives for multicomponent frameworks, starting with the general group–supergroup construction in Baburin (2016 ▸) for transitive nets. This method underlies various algorithms for construction and enumeration. In this direction we also define maximal symmetry periodic isotopes in terms of extremal group–supergroup indices of the components. Finally, turning towards generically, or randomly, nested components, we indicate the role of Burnside’s lemma in counting all periodic isotopes for classes of shift-homogeneous nets.

We next determine the number of periodic isotopy types of various families of embedded nets (linear 3-periodic nets) in whose components are embeddings of the net pcu. The simplest family here consists of those nets with n parallel components, each being a shifted copy of the model net . In this case we refer to as a multigrid or n-grid. Such nets have dimension type {3; 3} and are shift-homogeneous.

We give a brief indication of research directions in the determination of periodic isotopy classes and periodic isotopes for embedded nets with a double-vertex quotient graph as well as research directions in rigidity and flexibility.




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