# A topological method for the classification of entanglements in crystal networks

## Summary (3 min read)

### 1. Introduction

- Batten & Robson (1998) introduced the terms inclined and parallel interpenetration for different methods of interlocking two-periodic networks.
- A true topological description of interpenetration should be free of crystallographic symmetry relationships which may be affected by non-topological factors such as molecular geometries, the presence and placement of guest species etc.
- 1 electronic reprint A more detailed classification of all homogeneous twoperiodic and three-periodic interpenetrating sphere packings in cubic, hexagonal and tetragonal crystal systems was developed by Fischer & Koch (1976) and Koch et al. (2006).

### 2. The method

- Let us restrict their consideration to the entanglements caused only by Hopf links and/or multiple crossing links [as observed in interpenetrated quartz networks (Delgado-Friedrichs et al., 2003)] between the network cycles.
- If the authors then represent each ring by its barycentre and connect the barycentres of catenating rings they obtain what they call the Hopf ring net (HRN), i.e. the net whose nodes and edges correspond to rings and Hopf links between them.
- Therefore the authors can distinguish edges in the CRN according to the type assigned, which corresponds to different methods of Acta Cryst. (2012).
- With such different descriptions for the kind of edges,2 the CRN of the entanglement can be split into two subnets: a partial ring net, which is derived from the links between the barycentres of the rings belonging to the same network, and the HRN.
- Resuming, before comparing two HRNs (i.e. two catenation patterns), they should be pruned of collisions and the nodes corresponding to inessential rings.

### 3. Examples

- Below the authors consider different kinds of network arrays and methods of entanglement to demonstrate the applicability of the HRN approach.
- To designate nets, the authors use bold three-letter RCSR symbols (e.g. dia for the diamondoid network) and the Fischer k/m/fn nomenclature for sphere packings (Koch et al., 2006).

### 3.1. Networks admitting naturally self-dual tilings

- Most naturally, twofold interpenetration is realized in networks that admit a self-dual tiling: the networks are isomorphic to their duals (Delgado-Friedrichs et al., 2007).
- The crystallographic data on the interpenetrated arrays for cds, dia, hms, pcu, pyr, sda, srs and tfa were taken from the RCSR database (reported with the suffix -c).
- In the hms-c array each of the two non-equivalent 6-rings is catenated by six other rings (Fig. 5, middle); they correspond to the first coordination shell of the (6,6)-coordinated HRN (Table 2).
- Obviously, such subtle differences can hardly be revealed by a visual analysis.
- One feature of the interpenetrating arrays under consideration follows from the property of naturally self-dual networks: every essential ring of one network is crossed by one and only one edge of the other network, i.e. a twofold array of self-dual networks is fully catenated (all essential rings are catenated).

### 3.2. Interpenetrating sphere packings and coordination networks

- A good test to verify the proposed approach is to compare the catenation patterns in terms of simplified HRNs with the interpenetration patterns of sphere packings by Koch et al. (2006).
- Almost each type of interpenetration (a–r) matches a distinct catenation pattern represented by a unique simplified HRN.
- The most typical interpenetration for twoperiodic coordination networks, a twofold array of square (sql) networks, has the HRNs of the same sql topology if every 4-ring of one network is crossed by four 4-rings of the other network as in [( 2-5-(2-(3-pyridyl)ethenyl)thiophene-2carboxylato)2Zn].

### 3.3. Polycatenation

- As was mentioned above, the polycatenation phenomenon features the arrays where the periodicities m, n of interweaving networks are less than the periodicity k of the whole array.
- This is the case of interpenetrating two-periodic hcb (63) and fes (4.82) layers of spheres (Koch et al., 2006) that all have the same chain-like HRNs since each catenated ring (6-ring in hcb and 8-ring in fes) is linked to two similar rings of the other layer (Fig. 9).
- The parallel type of polycatenation also admits various topologies of the corresponding HRNs.
- For the latter example, as for the twofold sql ACUCIK Acta Cryst. (2012).
- A68, 484–493 Eugeny V. Alexandrov et al. Classification of entanglements 491 electronic reprint (see Fig. 8), the polycatenation is realized by the presence of bent ligands represented by the additional 2-coordinated nodes on two opposite sides of the square.

### 3.4. Self-catenation

- Self-catenated nets are single nets that exhibit the peculiar feature of containing rings through which pass other components of the same network.
- The only problem is that not all self-catenated networks admit natural tilings; moreover, if the natural tiling exists, the catenated rings are always inessential and do not belong to the ring basis.
- This means that classification of catenation patterns can be difficult in some complicated cases: the ring basis should be chosen separately for the catenated rings.
- A68, 484–493 electronic reprint other network in the twofold array, while in the self-catenation the 12-rings of only one kind participate (Fig. 13).

### 4. Conclusion

- The Hopf ring nets (HRNs) are shown to be a rigorous method for identifying catenation patterns irrespective of the geometry of interweaving networks.
- There are around 40 structures described as dia fourfold interpenetrated subdivided into five different classes that are described with only two different HRNs.
- Such taxonomy should help us gain a deeper insight into the nature of entanglement and to develop the design methods for new interlocking motifs.
- Such multi-ring links can be represented as additional multi-coordinated nodes in the resulting ring net.
- VAB is grateful for the 2009/2010 Fellowship from Cariplo Foundation and Landau Network – Centro Volta (Como, Italy).

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...…of interpenetrating networks both in inorganic and organic crystals (such as cuprite Cu2O or -quinol) was done by Wells (1954); however, a long time passed before the investigations of chemically unbonded but spatially non-separable motifs became important to chemists (Batten & Robson, 1998)....

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...Entanglement in crystal structures is a fascinating phenomenon that has been intensively investigated since the 1990s when Robson and co-workers (Hoskins & Robson, 1990; Batten & Robson, 1998) drew attention to this part of crystal chemistry....

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...1), which rarely occur in crystal structures (Carlucci et al., 2003)....

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...Carlucci et al. (2003) also proposed two kinds of additional topological parameters: degree of catenation (Doc, that is the number of networks entangled to a particular one) and index of separation (Is, that is the number of networks that should be removed to disjoint the array into two unconnected…...

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...Carlucci et al. (2003) and Proserpio (2010) distinguished three types of entanglements: interpenetration, when m, n coincide with k; polycatenation, when m, n < k; and self-catenation (other equivalent terms are self-penetration or polyknotting, cf. Jensen et al., 2000; Ke et al., 2011)....

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### "A topological method for the classi..." refers background in this paper

...Entanglement in crystal structures is a fascinating phenomenon that has been intensively investigated since the 1990s when Robson and co-workers (Hoskins & Robson, 1990; Batten & Robson, 1998) drew attention to this part of crystal chemistry....

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### "A topological method for the classi..." refers background in this paper

...In more detail, we must refer to the topological classification of nets, represented by their vertex symbols (Blatov et al., 2010); if one of the ‘shortest rings’ is catenated by other ‘shortest rings’ of the same net we can speak of a ‘true’ case of self-penetration....

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