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Acta Crystallographica Section A
Foundations of
Crystallography
ISSN 0108-7673
Editor: W. Steurer
A topological method for the classification of entanglements
in crystal networks
Eugeny V. Alexandrov, Vladislav A. Blatov and Davide M. Proserpio
Acta Cryst.
(2012). A68, 484–493
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Acta Cryst.
(2012). A68, 484–493 Eugeny V. Alexandrov
et al.
· Classification of entanglements
research papers
484 doi:10.1107/S0108767312019034 Acta Cryst. (2012). A68, 484–493
Acta Crystallographica Section A
Foundations of
Crystallography
ISSN 0108-7673
Received 12 February 2012
Accepted 27 April 2012
# 2012 International Union of Crystallography
Printed in Singapore – all rights reserved
A topological method for the classification of
entanglements in crystal networks
1
Eugeny V. Alexandrov,
a
Vladislav A. Blatov
a
* and Davide M. Proserpio
b
*
a
Samara State University, Ac. Pavlov St. 1, Samara, 443011, Russian Federation, and
b
Dipartimento
di Chimica Strutturale e Stereochimica Inorganica (DCSSI), Universita
`
degli Studi di Milano, Via G.
Venezian 21, 20133, Milano, Italy. Correspondence e-mail: blatov@samsu.ru,
davide.proserpio@unimi.it
A rigorous method is proposed to describe and classify the topology of
entanglements between periodic networks if the links are of the Hopf type. The
catenation pattern is unambiguously identified by a net of barycentres of
catenating rings with edges corresponding to the Hopf links; this net is called
the Hopf ring net. 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 modelled network arrays and in coordination
networks are considered.
1. Introduction
Entanglement in crystal structures is a fascinating phenom-
enon 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. To the best of our knowledge, the first review of
interpenetrating networks both in inorganic and organic
crystals (such as cuprite Cu
2
Oor-quinol) was done by Wells
(1954); howev er, a long time passed before the investigations
of chemically unbonded but spatially non-separable motifs
became important to chemists (Batten & Robson, 1998). Now
the number of examples of merely three-dimensional inter-
penetration is almost 1000 (Alexandrov et al., 2011), while
other types of entanglements have not been comprehensively
catalogued.
The topological properties of entangled motifs were in focus
from the very beginning. It was clear that one of the important
properties was the periodicity of interlocking networks as
well as of the resulting whole entangled array. Wells (1954)
described all kinds of entanglements between three-periodic
networks known at that time. Batten & Robso n (1998)
introduced the terms inclined and parallel interpenetration for
different methods of interlocking two-periodic networks.
Batten (2001, 2010) proposed to describe entanglements with
the formula nD/mD ! kD (now the form nD+mD ! kDis
more useful) where m, n are periodicities of the entangled
networks, and k is the periodicity of the whole array. Carlucci
et al. (2003) and Proserpio (2010) distinguished three types of
entanglements: interpenetration, when m, n coincide with k;
polycatenation,whenm, n < k; and self-catenation (other
equivalent terms are self-penetration or polyknotting, cf.
Jensen et al., 2000; Ke et al., 2011). A self-catenated network
exhibits the pe culiar feature of containing rings through which
pass other components of the same network (see below for
details). 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 separatio n (Is, that is the number of networks that
should be removed to disjoint the array into two unconnected
parts) that allowed them to classify the entanglements more
thoroughly.
The first attempt to algorithmize the classification of
entanglements was undertaken by Blatov et al. (2004), who
developed a rigorous computer procedure to characterize
three-dimensional interpenetration by the degree of inter-
penetration (the number of entangled networks) and a number
of space-symmetry parameters. This procedure was used to
catalogue 301 interpenetrated metal–organic frameworks
from the Cambridge Structural Database (Allen, 2002). This
approach has also been applied for inorganic and hydrogen-
bonded networks and is routinely used by experimentalists to
analyse three-periodic interpenetration (Baburin et al., 2005,
2008a,b
). However, the classification criteria of Blatov et al.
(2004) are mainly geometrical and do not concern the details
of interlocking the entangled motifs. They introduce classes of
interpenetration in terms of crystallographic symmetry rela-
tionships which do not recognize topologically different
interpenetrating network arrays that have the same sets of the
space-symmetry param eters (i.e. for a given net, in the same
class of interpenetration there may be different entangle-
ments). A true topological descrip tion 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
A preliminary account of this work was presented at the workshop
‘Topological dynamics in physics and biology’ held in Pisa, 12–13 July 2011.
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A more detailed classification of all homogeneous two-
periodic and three-periodic interpenetrating sphere packings
in cubic, hexagonal and tetragonal crystal systems was devel-
oped by Fischer & Koch (1976) and Koch et al. (2006). They
proposed the term interpenetration pattern to distinguish
different methods of interlocking ne tworks irrespective of the
network parts (rings) not participating in the entanglement
and of the size of the entangled parts. All interpenetrati on
patterns of the homogeneous sphere packings were tabulated,
but practical application of the approach was hindered
because no clear parameters of the patterns were proposed
that would allow one to detect them in crystal structures.
In this paper, we propose a rigorous method to characterize
entanglements. The method is algorithmized and implemented
as a computer routine that makes it useful for distinguishing
and classifying entanglements of any complexity in periodic
network arrays.
2. The method
Let us restrict our consideration to the entan glements 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. As both Hopf and multiple
crossing links are pairwise (they occur between two rings, Fig.
1), below we refer always to Hopf links meaning multiple
crossing links too, if not otherwise specified. Thus we avoid,
for the present, the multi-ring Brunnian interlockings
including the Borromean entanglement (Fig. 1), which rarely
occur in crystal structures (Carlucci et al., 2003). We will use
the term catenation for the Hopf link as this method of
interlocking characterizes the class of organic molecules
catenanes. Further, we consider the Hopf links only between
strong rings, i.e. cycles that cannot be represented as sums of
smaller cycles (Delgado-Friedrichs & O’Keeff e, 2005). Unlike
cycles, the number of which is in general infinite, the set of
symmetry non-equivalent strong rings is always finite
(Goetzke, 1993). Moreover, strong rings characterize all
smallest windows in the network (Blatov et al., 2007) and
contain a cycle basis (any cycle in the network is either a
strong ring or a sum of strong rings), so they provide sufficient
information required to describe the overall catenation. Below
we will call them rings for short.
If we then represent each ring by its barycentre and connect
the barycentres of cate nating rings we
obtain what we call the Hopf ring net
(HRN), i.e. the net whose nodes and
edges correspond to rings and Hopf
links between them. The following
properties of the HRN are worth
mentioning: (i) the degree (coordina-
tion number) of a node is equal to the
number of rings catenating a particular
ring; (ii) some nodes can have the same
coordinates if the barycentres of the
corresponding rings coincide (i.e. the
nodes collide); (iii) the star of a node
defines the bouquet of catenating rings (by bouquet we mean
the union of a particular ring and all rings that catenate this
selected one), which can be considered as the smallest
collection of catenated rings that characterize the entangle-
ment (Fig. 2).
The concept of the Hopf ring net extends the notion of
complete ring net (CRN) introduced by Baburin & Blatov
(2007) for single nets. In general, the CRN is derived from a
particular network in the following way: the nodes of the CRN
correspond to the barycentres of all rings in the initial
network, while the edges of the CRN connect the rings that
are in contact in the initial network. The method of CRN
construction depends on the definition of such contacts (i.e.
edges). Baburin & Blatov (2007) treated the rings as
connected if they had common edges with the initial network
reference ring. In this work we extend the notion of CRN to
include the presence of Hopf links either from entanglement
of different networks and/or from self-catenation, adding also
the edges for the catenation of the Hopf links but assigning
them to a different type.
Therefore we can distinguish edges in the CRN according to
the type assigned, which corresponds to different methods of
Acta Cryst. (2012). A68, 484–493 Eugeny V. Alexandrov et al.
Classification of entanglements 485
research papers
Figure 2
Two interpenetrating primitive cubic (pcu) networks (array pcu-c) shown in red and blue as well as
the corresponding HRN of nbo (NbO) type highlighted in green. The bouquet of catenating rings
and the corresponding HRN star (green balls) are shown in the second picture from the left.
Figure 1
Hopf, multiple crossing and the three simplest three-component links.
The corresponding edges of the ring nets that connect the ring-net nodes
are shown by arrows. For the Borromean link, the ring-net fragment
contains an additional node in the centre of the link. The program
Knotplot (R. G. Scharein; http://www.knotplot.com/) was used to draw the
link pictures.
electronic reprint
linking rings. In this paper, we distinguish two kinds of edges
in the CRN: one corresponds to Hopf links and the other
conforms to any connection of rings within the same network
through the network vertices and edges (such connection may
be provided by one common edge, or just a common vertex, or
even an edge chain of any lengt h). A special case arises for a
self-catenated network where Hopf links exist between the
rings of the same network (see below).
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 betw een the
barycentres of the rings belonging to the same network, and
the HRN. This representation is useful to explore the peri-
odicity of entanglement between the networks: we can
determine the periodicities n and m of two catenating
networks (n and m are equal to the periodicities of the
corresponding partial ring nets where Hopf links are ignored)
as well as the periodicity k of the CRN (i.e. the periodicity that
includes the presence of both kinds of links between rings);
the method of entanglement can then be written as nD+mD
! kD. It must be noted that the periodicity h of the HRN is
limited by k (h k), but can be higher than n or m (see
examples of polycatenation below).
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. For example, if a
set of structures containing two interpenetrating diamondoid
networks is under consideration, it is sufficient to compare
their HRNs to find the differences in their catenation patterns.
However, to match catenation patterns of topologically
different interpenetrating arrays (the number of inter-
penetrating networks must be fixed anyway) their HRNs
should be reduced beforehand with the two simplification
procedures described below.
First, in general, not all rings are catenated, and hence not
all of them are represented in the HRN. If a catenated ring A
is a sum of another catenated ring B and any number of non-
catenated rings, then A and B are equally catenated and can
be rep laced with a single ring in the HRN. This operation is
equivalent to fusing the HRN nodes corresponding to A and
B. Indeed, such equally catenated rings do not carry any new
information about the entanglement and can be treated as the
same ring in the catenation pattern. In the HRN, suc h rings
can be detected as collisions (Delgado-Friedrichs & O’Keeffe,
2005); the corresponding nodes have the same set of neigh-
bours. For instance, three distinct types of 14-rings in dia-f
(that is a decorated version of the diamond network) are
almost coincident (they share 12 vertices of the 14; see Fig. 3,
left). As a result they have three distinct but almost coincident
barycentres (the three green spheres) givin g rise to three
superimposed stars. Because each 14-ring is catenated by the
same set of 18 (six triplets) other 14-rings and share the same
stars in the HRN, the corresponding triplets of the HRN nodes
collide (Fig. 3, middle). After removing collisions each triplet
collapses into one node 6-coordinated (18/3 = 6) resulting in
an HRN of the hxg topology (Fig. 3, right).
Second, not all remaining rings in each interpenetrating
network are independent; some of them are still sums of
several catenated rings. Obviously, such dependent rings also
do not carry any new information on the catenation. So one
should consider only rings from the ring basis, i.e. the minimal
set of rings that generate other rings by summation. To the
best of our knowledge, there is no general algorithm to
determine the ring basis for a periodic network, but a partial
and useful solution of this problem may be found within the
natural tiling approach. According to Blatov et al. (2007) the
natural tiling is the unique method to represent a network as a
set of cages (natural tiles) that are confined by strong rings of a
special kind called essential rings. An important property of an
essential ring is that any strong ring is either an essential ring
or a sum of essential rings, i.e. the set of essential rings can be
treated as a ring basis. The problem is that there are networks
that do not admit the natural tiling, but they are mostly
unimportant for crystal chemistry because of their rare
occurrence. Indeed, all abundant three-dimensional under-
lying networks in inorganic, organic or metal–organic
compounds (see Alexandrov et al., 2011 and references
therein) admit natural tilings [for tilings of networks see the
Reticular Chemistry Structure Resource (RCSR), http://
rcsr.anu.edu.au/; O’Keeffe et al., 2008].
Notwithstanding the fact that the method is applicable to
any set of given rings, so it could also be used to analyse
entanglements for nets that do not admit natural tilings. All
the networks considered in this paper, except specially
discussed examples of self-catenation, admit the natural tiling,
so we will use this approach to determine the ring basis.
Resuming, before compari ng two HRNs (i.e. two catenation
patterns), they should be pruned of collisions and the nodes
corresponding to inessential rings. The HRN simplified in this
way unambiguously determines the topology of the catenation
pattern if the ambient isotopy is not taken into account. Recall
that ambient isotopic networks can be superposed in the space
without breaking edges or rings (Hyde & Delgado-Friedrichs,
2011). For example, 3-chain and (3,3)-torus links (Fig. 1) are
not ambient isotopic but give rise to equiva lent HRN frag-
ments. One can state that two entangled arrays of networks
research papers
486 Eugeny V. Alexandrov et al.
Classification of entanglements Acta Cryst. (2012). A68, 484–493
Figure 3
Three equally catenated 14-rings in an interpenetrating array of two
decorated diamondoid (dia-f) networks and the corresponding fragment
of the simplified HRN. All the 14-rings are related via non-catenated
4-rings: 14a-ring = 14b-ring + 4-ring = 14c-ring + 4-ring + 4-ring (see text
for details).
2
In practice, TOPOS (Blatov, 2006; http://www.topos.samsu.ru) assigns to
‘Valence’ the edges between the ring barycentres and to ‘Hydrogen bond’ the
Hopf link edges.
electronic reprint
have the same catenation patterns if and only if they have
isomorphic simplified HRNs. The isomorphism of HRNs can
be checked with the methods developed for periodic nets
(Blatov, 2007); if the HRN is a finite graph, the corresponding
methods from graph theory should be used. The TOPOS TTD
collection (Blatov & Proserpio, 2009) can be used to assign the
name to the net topology; this collection currently contains
more than 72 000 net topologies. Thus the method is purely
topological; the catenation patterns can be classified irre-
spective of the space-group symmetry and geometrical
embedding of the entangled networks. It is important that the
HRN topology i s independent of the size of catenating rings as
well as of the number and size of non-catenating rings in the
entangled networks. For example, the extension of edges or
the decoration of nodes (without the addition of new Hopf
links) in the networks does not influence the resulting HRN.
Fig. 4 illustrates the similarity of interpenetration in nine
different twofold network arrays (eight of them are inter-
penetrating sphere packings, see below). Here we consider
some possible interpenetrations; the detailed analysis of all
known catenation patterns for a given set of interpenetrating
networks is in progress and will be discussed elsewhere. The
local similarity is obvious from their bouquet s, which lead to
the same star of five ring barycentres (cf. Fig. 2). Only one
interpenetrating array which consists of two pcu networks is
fully catenated, i.e. has all rings catenated; in all other cases
there are either inessential rings (cab, nbo) or some essential
rings are not catenated (afw, cab, nbo-a, pcu-g, pcu-h, uku,
unp). Only in two cases (uku, unp) are there links that lead to
collisions to be removed (Table 1). Nonetheless, the simplifi-
cation procedure, i.e. removing both collisions and the nodes
corresponding to inessential rings, gives rise to the same
simplified HRN nbo for all the arrays, which proves that ther e
is the same catenation pattern in all cases.
We have algorithmized the method and impleme nted it in
the program package TOPOS (Blatov, 2006; http://www.
topos.samsu.ru). TOPOS provides the analysis of catenation
patterns according to the following general algorithm:
(i) searching for all cycles in the netw ork array and selecting
strong rings [see Blatov (2006) for details];
(ii) searching for all entanglements and selecting Hopf links
[see Blatov (2006) for details];
(iii) distinguishing Hopf links between different networks
(catenation) and within the same network (self-catenation);
(iv) constructing an HRN;
(v) removing collisions from the HRN (merging the nodes);
(vi) determining essential rings in accordance with the
algorithm by Blatov et al. (2007) and removing all HRN nodes
that correspond to inessential rings;
(vii) determining the HRN topology according to Blatov
(2007).
3. Examples
Below we consider differe nt kinds of network arrays and
methods of entanglement to demonstrate the applicability of
the HRN approach. To designate nets, we 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). By
definition, in the naturally dual network, the nodes, edges,
essential rings and natural tiles are in one-to-one relation to
natural tiles, essential rings, edges and nodes of the network
under consideration. The simplified HRNs (catenation
patterns) for twofold arrays of 23 interpenetrating naturally
self-dual networks which are stored in the RCSR database are
given in Table 2. The crystallographic data on the inter-
penetrated arrays for cds, dia, hms, pcu, pyr, sda, srs and tfa
were taken from the RCSR database (reported with the suffix
-c). For the other self-dual networks the interpenetrated
arrays have not yet been found in crystal structures (with
the exception of sxd); no data on their embeddings are
available so far. In this case we have generated an embedding
using the TOPO S procedure for constructing dual nets
(Blatov et al., 2007). For one twofold array of naturally self-
dual networks [(4,4)-coor bbr] we could not find any faithful
Acta Cryst. (2012). A68, 484–493 Eugeny V. Alexandrov et al.
Classification of entanglements 487
research papers
Figure 4
The bouquets of catenating rings in nine different network arrays that
lead to the same star of simplified HRN of the nbo topology (cf. Fig. 2).
Table 1
Strong and essential rings in twofold interpenetrating arrays of some
networks.
Network
Sizes of
strong rings
Sizes of
essential rings
Sizes of catenated
essential rings
pcu†4 4 4
nbo 6, 8 6 6
afw (pcu-n)3,7 3,7 7
cab (pcu-a) 3,4,8 3,8 8
unp (pcu-p) 3,4,9 3,4,9 9
pcu-g 6, 10 6, 10 10
pcu-h† 6,10 6,10 10
uku 3, 4, 6, 10 3, 4, 6, 10 10
nbo-a† 4,12 4,12 12
† Array is observed in real structures.
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