A topological method for classification of entanglements in crystal networks
Eugeny V. Alexandrov, Vladislav A. Blatov, Davide Maria Proserpio · Acta Crystallographica Section A Foundations of Crystallography · 2012
A rigorous method is proposed to describe and classify topology of entanglements between periodic networks.We consider the entanglements caused by Hopf links and/or multiple crossing links [1] between strong rings, i.e. cycles that cannot be represented as sums of smaller cycles [2].If then each ring is represented by its barycenter and the barycenters of catenating rings are interconnected we obtain a Hopf ring net (HRN).The HRN directly characterizes the catenation pattern, i.e. the method of catenation of the rings, if the kind of network and the degree of interpenetration are fixed.Thus, the topological type of HRN can be considered as the basic taxon for classification of entanglements.Before comparing two HRNs (i.e. two catenation patterns), they should be simplified, i.e. pruned of collisions [2] and the nodes corresponding to inessential rings (i.e. the rings that do not belong to the ring basis) [3].We have algorithmized the method and implemented it into the program package TOPOS [4].The Hopf ring net approach is compared with other methods of characterizing entanglements; a number of applications of this approach to various kinds of entanglement (interpenetration, polycatenation, and self-catenation) both in modeled network arrays and in coordination polymers are considered.In particular, we found that all interpenetration patterns in 149 three-periodic sphere packings [5] are matched to 18 simplified HRNs.We showed that the approach can easily be extended to other types of links between rings, such as the Borromean entanglement or, more general, the Brunnian interlocking.The comprehensive taxonomy of entanglements in coordination polymers should help us to develop design methods for new interlocking motifs.