On the group-theoretical approach to the study of interpenetrating nets

Abstract

Using group-subgroup and group-supergroup relations, a general theoretical framework is developed to describe and derive interpenetrating 3-periodic nets. The generation of interpenetration patterns is readily accomplished by replicating a single net with a supergroup G of its space group H under the condition that site symmetries of vertices and edges are the same in both H and G. It is shown that interpenetrating nets cannot be mapped onto each other by mirror reflections because otherwise edge crossings would necessarily occur in the embedding. For the same reason any other rotation or roto-inversion axes from G \ H are not allowed to intersect vertices or edges of the nets. This property significantly narrows the set of supergroups to be included in the derivation of interpenetrating nets. A procedure is described based on the automorphism group of a Hopf ring net [Alexandrov et al. (2012). Acta Cryst. A68, 484-493] to determine maximal symmetries compatible with interpenetration patterns. The proposed approach is illustrated by examples of twofold interpenetrated utp, dia and pcu nets, as well as multiple copies of enantiomorphic quartz (qtz) networks. Some applications to polycatenated 2-periodic layers are also discussed.

Isotopy classes for 3-periodic net embeddings

“…Also, for any subnet M the image g Á M is connected if and only if M is connected, and so the lemma follows. & These lemmas feature in the proof of the following theorem (Baburin, 2016).…”

“…We now give some useful group-theoretic perspectives for multicomponent frameworks, starting with the general groupsupergroup construction in Baburin (2016) for transitive nets. This method underlies various algorithms for construction and enumeration.…”

“…Such component transitive periodic nets have been considered in detail by Baburin (2016) with regard to their construction through group-supergroup methods.…”

“…Their classification and analysis in terms of symmetry, geometry and topological connectivity is an ongoing research direction (Batten & Robson, 1998;Carlucci et al, 2003Carlucci et al, , 2014Blatov 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).…”

“…Also we define periodic isotopy equivalence for embedded nets and prove that it is an equivalence relation and that there are finitely many periodic isotopes with a common labelled quotient graph. In the group methods of Section 7 we give the group-supergroup construction of entangled nets (Baburin, 2016), the definition of maximal symmetry periodic isotopes, and the role of Burnside's lemma in counting periodic isotopes. In Section 8 we determine periodic isotopy classes and also restricted periodic isotopy classes for various multicomponent shift homogeneous embeddings of n-fold pcu.…”