## Abstract

Three-dimensional entanglements, including knots, knotted graphs, periodic arrays of woven filaments and interpenetrating nets, form an integral part of structure analysis because they influence various physical properties. Ideal embeddings of these entanglements give insight into identification and classification of the geometry and physically relevant configurations *in vivo*. This paper introduces an algorithm for the tightening of finite, periodic and branched entanglements to a least energy form. Our algorithm draws inspiration from the Shrink-On-No-Overlaps (SONO) (Pieranski 1998 In *Ideal knots* (eds A Stasiak, V Katritch, LH Kauffman), vol. 19, pp. 20–41.) algorithm for the tightening of knots and links: we call it Periodic-Branched Shrink-On-No-Overlaps (PB-SONO). We reproduce published results for ideal configurations of knots using PB-SONO. We then examine ideal geometry for finite entangled graphs, including *θ*-graphs and entangled tetrahedron- and cube-graphs. Finally, we compute ideal conformations of periodic weavings and entangled nets. The resulting ideal geometry is intriguing: we see spontaneous symmetrisation in some cases, breaking of symmetry in others, as well as configurations reminiscent of biological and chemical structures in nature.

## 1. Introduction

The nexus between structure and function is a long-recognized concept in materials science, going back at least as far as Johannes Kepler, whose dictum *ubi materia, ibi geometria* [1], remains as relevant today as when it was penned in 1601. One measure of geometry in molecular materials, that is less resolved than the detailed coordinate geometry, but captures structural detail beyond pure topology, is that of *entanglement*, or knotting. Organic chemists have long pursued the synthesis of knotted synthetic molecules in response to their connections to molecular chirality as much as their intrinsic beauty [2–7]. The electrophoretic mobility of circular DNA has been shown to be sensitive to its knottedness [8], where synthetic design of knotted DNA and RNA complexes with prescribed entanglement is now possible [9–11]. In recent work, the more complex case of entangled finite graph embeddings has been investigated using similar approaches [4,12–18]. Replicating DNA often pass through configurations that are finite graphs [19].

Entanglements of finite loops (knots and links) have been explored from various perspectives within knot theory [20]. Generic networks—or graphs—are topologically more complex than loops, containing vertices common to multiple edges. Mathematical characterization of their possible entanglements is a complex and relatively unexplored area. To date, analysis has been mostly confined to finite entangled graphs [16,17,21,22]. The concept and description of entanglement in infinite graphs—particularly crystalline nets—is even more complex. The issue is an important one in understanding interpenetration of multiple networks [14,23–28] and weavings of infinite filaments [29]. Interpenetrating networks are ubiquitous in metal-organic frameworks (MOFs) and strongly affect their microporosity, a characteristic of central relevance to their efficacy as functional materials [30]. Characterization of catenation in MOFs has been the subject of a number of studies [24,31–33]. Interpenetrating frameworks are also found in bicontinuous liquid crystalline mesophases and related mesoporous tricontinuous inorganic derivatives [34–36]. The presence of distinct knottings of topologically equivalent nets has also been noted in MOFs [31]. The variety of distinct catenation modes of nets in porous frameworks, is likely to continue to grow, given the ever-expanding range of organic ligands and complexing agents being used in these materials. It is therefore of fundamental scientific interest to develop tools to classify, quantify and examine entanglements of periodic nets.

Established techniques such as analysis of nets by their constituent knots and links [37], minimal and average crossing numbers [38], and energy quantities for single filaments [39,40], provide interesting insights, yet are far from comprehensive. We examine this problem though an extension of the concept of *ideal knots* [20,41–43] to periodic entanglements. This approach has proven to be useful for characterizing finite knots and links [44–51]. In the case of electrophoretic mobility of DNA already mentioned, the mobility is found to also be sensitive to the average crossing number of the ideal form of the knotted loop, an indication that the physical shape of the knot is driving how quickly it moves [52,53]. In addition, energy minimizing conformations of two tangled strands are also seen in the shape of a DNA double helix, an important form of biological entanglement [54].

We define an ‘ideal’ geometry to be an embedding of an entangled graph which minimizes an energy functional. A number of alternative energy functionals for knots have been proposed and explored, including a *minimum distance* energy [55], *symmetric* energy [56,57] and *conformal* energy [58,59]. A comprehensive summary of energy measures for finite entanglements can be found in [60]; all have inherent advantages and disadvantages. Here we adopt an energy functional that is amenable to numerical exploration: a dimensionless knot tightening energy, where the energy (*ropelength*) of a given knot geometry is equal to the ratio of length of the constituent ‘rope’ to diameter [39,61,62]. The rope has excluded volume, defined by a tube of given radius. Ideally, one would deform the smooth continuous form in space avoiding edge crossings to arrive at an ideal embedding with minimal ropelength. In practice, this is more easily done on a discretized polygonal structure composed of vertices and straight edges [51,58,59,63,64]. It has been shown that polygonal ropelength minima exist for knots and that they converge to the minimal ropelength of the smooth knot type [65]. Analogous results are still lacking for ideal embedded graphs, and we suspect that edge branching at graph vertices makes these proofs more complicated. Furthermore, the existence of a unique geometry for an ideal configuration has been shown to be false for a number of specific links [61]. The minimal ropelengths are, however, probably unique. Computational experiments suggest that numerical realization of global minima for generic knots is complicated by the presence of deep local minima [66]. With these cautionary notes in mind, we have explored the more vexed issue of tight embeddings of graphs. We do not claim the last word in tight embeddings of the graphs analysed here; rather the paper presents one approach to the problem and presents associated results. The hope is that this will stimulate further work, which is surely needed before final conclusions can be drawn.

In this article, we extend the Shrink-On-No-Overlaps (SONO) [63] algorithm, which computes an ideal embedding of a knot, to a new Periodic-Branched Shrink-On-No-Overlaps (PB-SONO) algorithm, to find ideal configurations of entangled finite graphs, three-periodic weavings and periodic networks. The extension is similar to that developed for tight open knots [67–69], however, it goes further to support three-periodicity. As an initial test of the algorithm, it replicates tight knot embeddings, comparable to the SONO algorithm, and we present tight embeddings of entangled *θ*-, tetrahedron- and cube-graphs to demonstrate its efficacy at dealing with vertices in the structure. We then discuss ideal configurations for a number of three-periodic weavings, made of unbranched filaments and two- and three-periodic nets, and compare those results with alternative approaches to canonical net embeddings. Lastly, we show that this tightening procedure affords useful embeddings of multiple catenated three-periodic nets, allowing us to distinguish distinct catenations of topologically equivalent nets.

Simulations of polymer melts typically use periodic boundary conditions and generate complex periodic weavings that have been studied using topological constraint graphs and generalized definitions of linking numbers [70,71]. These linking numbers may be correlated to our periodic ropelength as is the case for finite knots and average crossing number [20].

Some clarification of our terminology is perhaps helpful before describing the results of our algorithm. Three distinct levels of structure will be explored here, topology, entanglement and geometry. The topology of a pattern is captured by its associated graph, without reference to possible tangles or specific forms of edges. Entanglement refers to the mutual threading and twisting of edges, and in practice the same topological structure, or graph, can be realized by distinct entanglements. We call distinct entanglements of the same topological structure *isotopes* [26]. The most detailed level of structural description is the specific geometric conformation, or *embedding* of the entangled graph in space, described for example by the spatial coordinates of edges and vertices of the structure in space.

## 2. Tightening branched and periodic entanglements

The SONO algorithm is an efficient numerical implementation of the ropelength functional for finite knots and links. It uses a repulsion mechanism to swell adjacent segments of the knot, coupled with a shrinking mechanism to tighten the knot. Mechanisms are in place within the algorithm to shake the knot out of local minimum energy conformations, but a proof that these mechanisms will always ensure a global minimum is reached is elusive and likely untrue in general. Despite the computational and approximate nature of the algorithm, SONO has proven to be effective in the analysis of ideal knots and links [63].

A full description of the SONO algorithm for finite knots and links can be found in [63], we give only a brief summary here. Initially, the starting configuration of the knot is discretized into a series of ‘nodes’ connected consecutively by ‘leashes’. The radius of the knot is simulated by placing spheres of the same radius at points along the knot trajectory, where the distance between the points (the leash length) is far smaller than the sphere radius. The algorithm shrinks the knot trajectory (by scaling all coordinates of nodes along the knot trajectory by a factor less than 1) while maintaining this radius. When the spheres surrounding points in distinct sections of the knot overlap, they repel until the overlap is removed and the shrinking proceeds, eventually giving the ideal configuration (minimizing ropelength, *L*/*D*).

The SONO algorithm is insufficient to tighten structures with branches (graph vertices) as the shrinking mechanism cannot induce local length minimization in the vicinity of graph vertices. SONO is also insufficient to tighten periodic structures; the fixed ratio of rope diameter to unit cell size in periodic structures prohibits ‘shrinking’ the structure as a means of tightening. These two obstacles to forming ideal embeddings can be overcome via the introduction of a new process, we term ‘Tension’. Here we detail our generalized algorithm, PB-SONO, applicable to (possibly infinite and periodic) branched knots (tangled graphs or nets) as well as unbranched knots.

### (a) Tension

The procedure ‘Tension’ tightens the structure and replaces that of scaling (shrinking) in SONO. This procedure is necessary for tightening both periodic entanglements and those with vertices. The tension process locally straightens each segment of the discretized geometry (nodes and leashes, as for the SONO algorithm). A node is chosen at random and moved a small step towards the midpoint of its neighbours, then subsequently applied to every node. Where the chosen node is a vertex (found in branched graphs or nets), the node is moved to the barycenter of the adjacent nodes. We prevent a node from moving towards its target if this would introduce an overlap with the sphere around another nodes: where the desired position of the chosen node is within *D* of any other nodes in the entanglement (excluding neighbouring nodes), the node remains in its initial position. In most cases, the tension process keeps the length of the leashes consistent. To ensure that the variation of leash lengths remains small in all other cases, we also delete or insert nodes where the leashes are too short or long, respectively.

### (b) Find neighbours and remove overlaps

The ‘Find neighbours and remove overlaps’ procedure is done mostly as per the SONO algorithm. This process ensures that distinct parts of the structure do not overlap. When the spheres surrounding two nodes from distinct parts of the structure overlap (or are very close), they are displaced symmetrically away from each other, each by a small distance, *s*. However, spheres from neighbouring nodes must be allowed to overlap in order to approximate a rope-like shape. An integer number of neighbouring nodes to either side of a given node are allowed to overlap. This number is selected based on a tightest u-turn that is allowed and is best approximated by the closest integer to (*πD*/2*l*), where *l* is the average leash length and *D* is the diameter.

In PB-SONO, we also consider vertices. Where nodes are on disjoint edges of the graph (i.e. those with no common vertex), they repel as normal. In the case where edges of the graph share a vertex, an integer number of nodes closest to that vertex are permitted to overlap. This number depends on a vertex angle (*α*), which describes the minimum angle permitted between edges at the vertex and is given by the integer closest to
*α* in the case of ‘multigraphs’ only, such as the untangled *θ*-graph example discussed later. (Multigraphs are graphs with more than one edge connecting the same pair of vertices). For all ‘simple’ graphs (with at most one edge connecting each pair of vertices, such as the tetrahedron or cube) repulsion between disjoint edges of the graph is sufficient to ensure that the ideal configuration found by PB-SONO is independent of *α*.

The added benefit of this procedure is that it helps the structure avoid local minima by setting the small distance that nodes are mutually displaced to be slightly higher than the sphere diameter. This encourages tangled sections to slide laterally where necessary.

### (c) Tightening process

The original SONO algorithm alternates between removing overlaps and shrinking the knot trajectory while maintaining the diameter. The PB-SONO algorithm requires that there is no net change in the ratio of filament diameter to unit cell size. Given a structure, either finite or within a periodic unit cell, and a starting radius, PB-SONO proceeds as follows:

(i) shrink filament diameter by a particular fraction (

*D*_{small}=*D*×*f*_{shrink});(ii) perform ‘tension’ process multiple times and remove overlaps using diameter

*D*;(iii) increase filament diameter back to

*D*.

For finite entanglements (knots, links and knotted graphs), sufficient repetition of the process will yield an ideal configuration. In general, halving the number of nodes in the discretization of the trajectory for several steps and subsequent doubling of the nodes, assists in escaping local minima where the structure may be stuck. For periodic entanglements, the procedure described above will converge to a configuration for a given filament diameter. The diameter is then incrementally inflated, and the algorithm repeated for each new diameter. The inflation is performed until the structure becomes jammed, which can be identified by the inability of the ‘remove overlaps’ procedure to reach a configuration without any overlaps (the spherical nodes vibrate between each other). This conformation minimizes *L*/*D* for periodic structures, both filaments and networks, given the fixed unit cell. The inflation of the filament diameter is necessary to avoid a dependence of the final configuration on the starting configuration.

### (d) Periodic unit cell

Periodic entanglements are defined by entangled components within a repeating unit cell. In general, it is necessary to search for tight conformations of periodic entanglements among all deformations of the unit cell size and shape; the ideal configuration is that with the least ropelength among all choices of lattice parameters. Deformation of the periodic unit cell is achieved by varying the distance metric between points with crystallographic coordinates {*x*,*y*,*z*} and {*p*,*q*,*r*}, subject to variable unit cell dimensions (*a*,*b*,*c*,*α*,*β*,*γ*):

For example, many of the periodic entanglements shown in this article have cubic symmetry and are tightened within a cubic unit cell (lattice parameters *a*=*b*=*c*=1; *α*=*β*=*γ*=*π*/2).

A full PB-SONO simulation run, as described under ‘Tightening Process’, should be performed for many distinct sets of lattice parameters, with the global minimum taken as the ideal configuration. However, our current procedure for altering lattice parameters is very time consuming, and some intuition is usually needed to accelerate convergence to the ideal embedding. This is a clear limitation of our current algorithm, and implementation of a more efficient method to sweep unit cell shapes is needed.

## 3. Tight knot conformations

A test for the efficacy of PB-SONO is to compare its results with those of SONO quoted in [20], for classical knots with up to seven crossings (table 1). For the simpler knots, PB-SONO performs reasonably when compared with SONO. PB-SONO is slower and seems to have more difficulty in escaping from local minima, but the process of halving and subsequently doubling the number of points described above is a useful strategy. PB-SONO has difficulties with the 6_{2} and 6_{3} knots, and the ropelengths obtained are somewhat higher than the best SONO result. Figure 1 shows each of the ideal conformations for these knots obtained by PB-SONO.

## 4. Tight conformations of finite graphs

### (a) *θ*-graphs

The simplest, non-trivial finite graph, the *θ*-graph, is composed of two vertices connected by three edges. Distinct isotopes of the *θ*-graph have been enumerated up to a given complexity initially by Litherland, and more completely by Moriuchi [21]. Illustrations of these isotopes are given in figure 2, listed by the Litherland name. We consider all embeddings up to 6_{4} (Litherland). We note that the 5_{1} and 6_{1} isotopes are ‘ravels’ [17,18]. In all cases, we have used initial configurations deduced from the planar images shown in figure 2. Ropelength data for the resulting ideal conformations of these graphs are shown in table 2 listed by the Litherland name as well as the Moriuchi name, which uses Conway's tangle notation [72]. They are ordered by increasing ropelength.

The *θ*-graph is topologically a multigraph (three edges connecting the same two vertices). The untangled isotope of the *θ*-graph thus requires a careful choice of the lower bound on the allowed vertex angle (as discussed in §2b). Where no lower bound is set, the tight configuration of the unknotted *θ*-graph adopts a vertex angle of 0°, and all three edges coincide with the line joining the vertices. Furthermore, all spheres in the discretized structure are allowed to overlap, so the graph degenerates to a single node. If we set the lower bound such that at least the three spheres positioned on the midpoints of each edge repel, the resulting ideal conformation of the unknotted *θ*-graph consists of three equivalent edges, lying along three semi-ellipses connecting the vertices. The principal diameters of the ellipses are *D* between the two vertices of the graph and *D*, and hence they must be displaced *θ*-graph.) This ideal conformation has *L*/*D*=5.09 and vertex angle 105°. This case—with just three sterically excluded spheres—lies between a stable tight configuration and the unstable collapsed form. In practice, it is difficult to achieve this embedding via simulation using PB-SONO; we find numerically that a stable tight embedding of the unknotted *θ*-graph has *L*/*D*=5.12, as shown in figure 3*a*,*b*. In contrast to the extreme sensitivity to allowed minimal vertex angle for this untangled isotope, all of the tangled *θ*-graph isotopes are insensitive to the imposed lower bound on the vertex angle as their edges are sufficiently tangled to prevent a collapse of the structure.

Approximate ropelengths of many tight knotted *θ*-graphs can be estimated from related knots. The 3_{1} *θ*-graph isotope is equivalent (by ambient isotopy) to a trefoil knot with a short additional edge; we expect its ropelength to be close to that of the ideal trefoil plus *D* units (the length of the extra edge), which translates to a ropelength of 16.38+1=17.38. In practice, we obtain a ropelength of 17.44 (figure 3*c*,*d*). We note that the ideal conformation of the 3_{1} *θ*-graph is not unique: the short connecting edge may be located at any point where different parts of the knot trajectory are in contact. Similarly, the 4_{1} *θ*-graph embedding is the 4_{1} (figure-eight) knot with an additional short edge. The ideal tight conformation, as shown in figure 4*a*,*d*, therefore has a ropelength of 21.72+1=22.72 (the ideal figure-eight knot plus one). In practice, our implementation of PB-SONO yields *L*/*D*=22.77. In this case too, the tight configuration is not unique, as the location of the short edge has some flexibility, giving many ideal configurations equal (minimal) length.

The 5_{2} *θ*-graph isotope has the next longest ropelength when tightened. This isotope contains a trefoil, with its additional edge wrapping around outside the trefoil. The presence of this edge necessarily distorts the ideal conformation away from the ideal trefoil embedding, as shown in figure 4*b*,*e*. Here, *L*/*D*=24.24. The 5_{3} *θ*-graph isotope contains a cinquefoil knot with an additional short edge. When tightened (figure 4*c*,*f*), *L*/*D*=24.79, comparable with the cinquefoil (23.67) plus one (for the edge). Slight deviations from the ideal knot trajectory (kinks) are seen at the vertices of the graph, which cause the difference in the values. These kinks are an artefact of the computational process.

The 5_{1} *θ*-graph is a ravel [17], with no knotted or linked cycles. This isotope is the simplest example of a ravel. (An organo-metallic molecule containing this specific entanglement has been synthesized [18].) The ideal conformation (figure 5*c*,*d*) has *L*/*D*=25.08 and exhibits threefold symmetry. Intermediate configurations of the simulation are shown in figure 5*a*,*b*; the symmetry of the ideal configuration emerges regardless of the asymmetry of the starting configuration.

The ideal conformations of the remaining *θ*-graph isotopes are shown in figure 6. Each of the 5_{6}, 5_{7} and 5_{5} *θ*-graphs contains the 5_{2} knot (figure 1*e*), with the addition of an edge between sections of the knot that are in contact and their tight embeddings are governed by tightening of the knot. As there are three topologically distinct contact locations, three isotopes emerge, all with distinct entanglements. If the resulting tight configurations have unperturbed (tight) 5_{2} knots, the resulting lengths for these three isotopes would be equivalent (*L*/*D*=25.05+1). In practice, PB-SONO generates slightly perturbed ideal configurations, with variable length (26.17, 26.30 and 26.53). Those variations are most probably because of limited numerical relaxation and we expect the lengths to converge with longer relaxation times. Note also that the additional edge has some flexibility in the 5_{6} and 5_{5} cases, allowing a suite of tight conformations with equal length. By contrast, the 5_{7} ideal embedding is unique, as the two segments of the knot containing the vertices are only in contact at a single point. The 6_{1} isotope is a ravel, likely to be second in length only to the 5_{1} isotope among all ravelled *θ*-graph embeddings.

These examples of tangled *θ*-graphs demonstrate the subtleties of tight embeddings. First, in some cases the embedding is not unique. Second, distinct entanglements can have equal normalized length in their tightest configurations. The latter issue is, however, rare, so that the normalized length generally offers a convenient ranking of the tangling complexity for topologically equivalent graphs.

### (b) Tetrahedron graphs

A tetrahedron graph consists of four degree-3 vertices. The untangled embedding of a tetrahedron embeds on the sphere without overlapping edges. The simplest entangled cases of a tetrahedron graph have embeddings in the torus, enumerated in detail elsewhere [22]. Using PB-SONO, we tighten the trivial tetrahedron graph plus five toroidal embeddings (figure 7), resulting in the rope lengths listed in table 3. We also show the ‘two-dimensional energy’ of the barycentric embedding within the flat torus, derived from the embedding of the related two-periodic graph in the universal cover of the flat torus, the two-dimensional Euclidean plane. The differences between this two-dimensional energy, and the ropelength which gauges a tight embedding in three-dimensional space and is therefore a three-dimensional energy measure, are described elsewhere [22,26].

The unknotted tetrahedron isotope with straight edges has a total length of approximately 6.93 diameters: this is achieved where the closest distance between opposite edges of the graph (those that do not share a common vertex) is *D*, hence each edge length must be approximately 1.16*D*. A tighter embedding, with *L*/*D*=6.61, is realized in a conformation with curved edges, as shown in figure 8*a*. This conformation is the ideal conformation of the trivial tetrahedron graph. The tetrahedron graph is a simple graph rather than a multigraph, and its ideal embedding is insensitive to an imposed lower bound on vertex angles, as disjoint edges repel each other. We also note that the curved geometry of the edges is not an artefact of a lower bound on the vertex angle, rather it is inherent to the geometry of the ideal tight structure.

The ideal isotope *A* (figure 8*b*) contains a knotted trefoil cycle. The complete isotope consists of this cycle with two additional short edges. The simulated ropelength is 18.54, comparable with the expected value of 16.38+2=18.38. A small kink is present at a vertex of this configuration, which we attribute to experimental uncertainty. The ideal isotope *B* is also dominated by a trefoil knotted cycle, with additional edges and vertices that induce a distinct entanglement of the tetrahedron graph. Its ropelength is 20.09 (figure 8*c*).

The *C* and *E* isotopes both contain knotted cinquefoil cycles (figure 8*d*,*e*, respectively). These two embeddings are distinct tetrahedron entanglements due to the additional edges: isotope *C* has two short connecting edges and isotope *E* contains a more complex structure. This is reflected in the ropelengths for these configurations, which are 25.80 for isotope *C* (2.13 more than the cinquefoil) and 27.42 for isotope *E* (3.85 more than the cinquefoil).

Among these six simplest tetrahedron isotopes, isotope *D* has the largest ropelength. This isotope contains a seven-crossing torus knot, evident in figure 8*f*. Its length is equal to that of the ideal torus knot, plus the additional two short connecting edges, which gives a ropelength of 33.22. If we rank entanglement complexity by ideal ropelength, all tangled isotopes are more complex than their untangled isotope, as expected, and the tetrahedron isotopes naturally order in the sequence *A*,*B*,*C*,*E*,*D*. Alternatively, we can rank the isotopes via their relaxed length within the flat two-dimensional torus, giving an effective length in their two-dimensional embedding [26]. The resulting two-dimensional energy for these graph embeddings, orders the isotopes as *A*,*B*,*C*,*D*,*E*, slightly different to our three-dimensional ropelength ranking (table 3). The broad agreement lends weight to the use of either measure, the minor differences confirms the impossibility of finding a single universal measure of entanglement complexity for even these simplest finite graphs.

### (c) Cube graphs

Five simple entanglements of the cube graph, isotopes *A* through *E*, are enumerated in [16]. A further five toroidal isotopes, 1 through 5, are enumerated in [22]. All 10 cube isotopes are shown in figure 9. The ropelengths for the ideal conformations of these structures are shown in table 4. As for the tetrahedron isotopes, their ranking by two-dimensional energies, as defined by their length in the flat torus determined elsewhere [22], differ from the three-dimensional energy, found here using PB-SONO.

The PB-SONO ideal embedding of a trivial cube graph contains straight edges, each of length *D*, with *L*/*D*=12.04, and it contains the complete set of isometries of the edge configuration of the Platonic cube (figure 10*a*,*d*).

Isotope *A* contains a Hopf link with the addition of four short edges joining the two components of the link. The ideal embedding of a bare Hopf link, has *L*/*D*=2*π*, i.e. *πD* for each component. In isotope *A* (figure 10*b*,*e*), the cycle around one component of the Hopf link consists of four edges. The minimum length for any edge in an ideal graph embedding is *D*, and as the cycle of four vertices must have length at least 4*D*, these cycles must deviate significantly from their ideal trajectory. According to PB-SONO, the tight isotope has ropelength 16.97.

Isotope *C* is the 3_{1} (trefoil) knot with four short connecting edges, giving an estimate on the ropelength of *L*/*D*=16.37+4=20.37. PB-SONO produces a tight isotope with ropelength 21.04 (figure 10*c*,*f*). We see that, compared with the *A* isotope of the tetrahedron graph, the addition of extra edges to form the cube isotope distorts the trefoil knot from its ideal conformation.

Figure 11 shows the ideal tight embeddings of the remaining cube isotopes. Isotope 1 contains a trefoil knot, and isotopes *B* and *D* (4,2) Whitehead links (where isotopes *B* and *D* differ only in the conformation of the additional edges attached to the (4,2) link).

Comparison of the energies realized by these tight embeddings of simpler cube isotopes in three-dimensional space with the relative energy rankings induced by the two-dimensional flat torus embeddings reveals some significant differences. In particular, isotopes 1 and 5 have relatively low three-dimensional ropelength energies, and high two-dimensional toroidal energies.

Comparison of the ideal embeddings of knots with more complex tetrahedron and cube isotopes containing those knots shows that, in many cases, the knotted cycles of ideal polyhedral isotopes adopt the ideal knot conformation, possibly slightly deformed because of the presence of additional edges and vertices. As a result, the ideal tangled polyhedra adopt symmetries carried over from those of the underlying ideal knots. We note that the most symmetric embeddings of tangled tetrahedron and cube isotopes, which are inevitably chiral [13], are of lower symmetry than their untangled (Platonic) versions. A separate analysis reveals that toroidal tetrahedron and cube isotopes can have, at most, rotational symmetry of order 2 and 4, respectively [22]. Within the inevitable uncertainty due to the PB-SONO algorithm, those symmetries are in fact manifested in the ideal embeddings.

## 5. Ideal weavings

A key motivation driving our extension of SONO is to explore ideal periodic *weavings*, including three-periodic rod packings [73] and packings of curved filaments [29]. The ideal conformation of a periodic weaving is defined to be the conformation that minimizes the ropelength, *L*/*D*, within a periodic unit cell of the structure. This ideal form may distort initially straight rods into curvilinear forms, and the tightening algorithm admits an (often unique) ideal (canonical) embedding for three-periodic weavings. (Ideal configurations of many weavings found via our PB-SONO algorithm are published in [29], however, we give a brief summary here.) As the magnitude of ropelength depends on the choice of unit cell volume, relative ranking of distinct weavings via this normalized ropelength is impossible. The volume fraction (or ‘packing fraction’, *viz.* *πLD*^{2}/4*V*) of the filament within a unit cell is a useful dimensionless measure of the conformation. Packing fraction is an intensive variable, independent of the unit cell chosen. We note in advance, that these alternative conformational ‘energies’ are inconsistent: minimal ropelength weavings do not correspond to those with maximal packing fraction.

### (a) Rod packings

From the perspective of entanglement, both rod packings and three-periodic weavings lie in the same class of patterns. Hitherto, rod packings were assumed to be built of straight cylinders; we prefer to extend the class to weavings of filaments, closed or open, and curvilinear or straight. We rename all such patterns ‘weavings’, to make this clear. Six invariant rod packings of cubic symmetry are enumerated in [29,73]. Table 5 summarizes the ropelengths (*L*/*D*) and packing fractions (*ρ*) of the ideal conformations of five of these packings, then shown in figure 12.

For all ideal embeddings of the cubic rod packings, except the ideal *Γ* rod packing, arbitrary deformations of the unit cell increase ropelength and decreases the packing fraction, Hence the cubic unit cell (with lattice parameters (*a*=*b*=*c*=1; *α*=*β*=*γ*=*π*/2) is optimal. For the ideal *Γ* structure, an increase in one unit cell edge, resulting in a tetragonal cell, gives a denser conformation, with a longer ropelength (larger *L*/*D*), and a higher packing fraction. If the shape of the tetragonal cell is fixed, the resulting tight (and non-ideal) configurations found by PB-SONO contain undulating rods, whose undulations increase with the tetragonal axial (*c*/*a*) ratio.

The achiral *Σ** packing, not given in the table or shown in figure 12, is an interweaving of the enantiomeric pair (*Σ*^{+} and *Σ*^{−}) of rod packings. The ideal embedding of the *Σ** rod packing (not shown) is exactly equivalent to the interweaving of the ideal enantiomeric structures: remarkably, the ideal *Σ*^{+} (or equivalently *Σ*^{−}) embeddings leave precisely the correct vacant space to intercalate the opposite enantiomer, also in its ideal form.

As shown in table 5, straight rods are present in the achiral packings (with the exception of the *Σ** packing), whereas chiral arrays induce quasi-helical rods. Straightening of helical rods, and consequent loosening of the tight packing, is accompanied by variations of packing fraction dependent on the particular packing. Those loosening modes are often auxetic (with negative Poisson's ratio), as explored in more detail elsewhere [74,75].

### (b) Complex weavings

A variety of more entangled weavings with cubic symmetry have been enumerated elsewhere [29], constructed as curvilinear filament trajectories on three-periodic minimal surfaces (TPMS). We have generated tight embeddings of these more complex weavings also using PB-SONO. The resulting ideal configurations are shown in figure 13.

It is worth noting that many examples retain the symmetry inherited from the underlying TPMS and have visually pleasing configurations, such as the structure in figure 13*d*. However, as with torus knots and links, tight embeddings may result in significant symmetry breaking. One example is shown in figure 14: the *P*432. Following tightening using PB-SONO, much of this symmetry is lost (figure 14). PB-SONO gives a ropelength, *L*/*D*=85.62, and packing fraction, *LD*^{2}/*V* =0.44. The tightening pathway is best described with reference to a local volume, which resembles a helix bundle composed of four mutually woven filaments. On tightening, one of the filaments straightens to occupy the centre of the helix bundle, and the other three undulate, forming a triple helix. A single loop of the weaving traverses four of these broken-symmetry tight helical bundles before closing up on itself, as shown in figure 14*b*. This symmetry breaking is reminiscent of the ideal configurations of high complexity torus knots and links, as well as the simple double helix, which lose symmetry on tightening [63]. This tightening pathway occurs locally throughout the weaving, where the filament that straightens is—according to the current implementation of PB-SONO—arbitrary. This leads to PB-SONO finding an arbitrary embedding. It does not exclude, however, the existence of a unique minimum, where the breaking of symmetry in a particular helix dictates a favourable configuration nearby, thus propagating throughout the structure. Finding this globally unique minimum is computationally sensitive and beyond the current implementation of PB-SONO.

## 6. Ideal configurations of periodic network entanglements

We turn our attention to ideal embeddings of two- and three-periodic nets and arrays of inter-woven nets. The ‘equilibrium placement’ concept of O'Keeffe & Delgado-Friedrichs [76] leads to canonical embeddings of crystallographic nets [77]. In fact, equilibrium placement allows a continuum of embeddings of a crystallographic net, as the unit cell parameters are arbitrary. However, two of those embeddings are most useful, and are explicitly computed by the SyStRe algorithm [78]. The first is a ‘barycentric’ embedding, which places each vertex at the barycentre of its neighbours. The second is a ‘uniform’ embedding, where the variance of the edge lengths is minimized, favouring uniform edge lengths. Despite the evident power of the SyStRe algorithm, there remain many nets for which it gives no canonical form, including those with edge and vertex collisions, where multiple edges or vertices occupy the same location in the equilibrium placement, and non-crystallographic nets. Additionally, SyStRe cannot produce canonical embeddings for patterns containing multiple interwoven nets. Given those limitations, it is worthwhile exploring ideal tight embeddings of nets via the PB-SONO algorithm. (Ideal configurations of many of these nets are published in [27].)

### (a) Simple periodic nets

Consider first the ideal configurations of the simplest three-periodic *regular* nets, that are edge-1 and vertex-1 transitive [79]: **srs**, **dia** and **pcu** [80]. On tightening, these nets are equivalent to both the barycentric and the uniform embeddings given by SyStRe. These ideal conformations, which realize all possible symmetries of their graph topologies, are shown in figure 15.

Consider next, ideal forms of two-periodic graphs. In its ideal form (figure 15*d*), the regular **hcb** (graphene) net has equal edges and ropelength *L*/*D*=2.89, with a hexagonal 2-periodic unit cell (*a*=*b*=1; *γ*=*π*/3), which is equivalent to both of the SyStRe embeddings. For the (4,4,8,8) two-periodic net, the barycentric embedding and the uniform embeddings given by SySrRe are distinct. The ideal embedding found by PB-SONO, shown in figure 15*e*, is equivalent to the uniform embedding and not the barycentric embedding. This ideal embedding has ropelength *L*/*D*=6.05, with lattice parameters *a*=*b*=1; *γ*=*π*/2.

### (b) Periodic nets with collisions

PB-SONO also allows us to form ideal embeddings of nets whose equilibrium placements (both barycentric and uniform) have collisions. An example is the 2(3,5)2 net, whose labelled quotient graph [81,82] is shown in figure 16*a*: two distinct vertices of this net occupy the same point according to SyStRe. An alternative embedding for this net was given in [83], where the vertices are at (0,0,0) and *b*, with lattice parameters (*a*=*b*=1; *α*=*β*=*γ*=*π*/2), and vertex positions (0,0,0) and (0.38,0.38,0), respectively. The steric restriction on the edge lengths (which are constrained to be at least *D* units, discussed above), ensures the vertices occupy distinct sites. For this ideal conformation, *L*/*D*=6.22.

### (c) Non-crystallographic nets

Consider next the non-crystallographic net whose quotient graph is shown in figure 17*a*. This net may be equivalently represented by multiple quotient graphs that are not isomorphic [83]. A possible embedding of this graph is given in [83]. The ideal configuration, shown in figure 17*b*,*c*, has lattice parameters (*a*=1; *α*=*β*=*γ*=*π*/2), and *L*/*D*=12.67. The embedding has vertex positions *A* *B* *C* *D* *E* *F* (0,0,0). Differences between this ideal embedding (given by PB-SONO) and the embedding described by Eon [83] are not large: the *x* coordinate of the *B* and *D* vertices is *not* the ideal unit cell) with lattice parameters *a*=1; *α*=*β*=*γ*=*π*/2, the locations of the vertices in both embeddings coincide.

### (d) Multiple catenated nets

A useful additional feature of ideal nets is the possibility of forming ideal embeddings for patterns containing multiple catenated disjoint nets [27] via tightening. Multiple inter-grown nets are frequently encountered in atomic and molecular materials [24,28,31] and are also of interest because of their photonic properties [84]. Consider, for example, structures containing two interpenetrating **srs** nets, both of equivalent chirality. Two catenations are known theoretically, with distinct entanglements [27]. On tightening (figure 18), we find that the *G*_{124RT}(cosh^{−1}(3)) structure has a significantly lower ropelength than the *G*_{129RT}(cosh^{−1}(5)) structure: *L*/*D* values are 28.1472 and 41.0965, respectively. This difference affords convincing evidence that they are not equivalent catenations, despite their equivalent topologies. Furthermore, the relative ranking of the two structures by ropelength implies that the *G*_{124RT}(cosh^{−1}(3)) structure is less entangled. Interestingly, the crystallographic data from a synthesized pair of **srs** with equivalent chirality given in [85] gives a conformation that matches the ideal form of the *G*_{124RT}(cosh^{−1}(3)) structure.

Catenations of four and eight **srs** nets have also been generated theoretically [23,27]. The ropelengths of ideal configurations of these patterns (figure 19) are 8.49 and 28.48, respectively. In both cases, the **srs** components adopt their ideal single **srs** formation, where all edges are straight and they are embedding with maximal (cubic) symmetries (one reported by [23]).

Ideal cubic configurations of two and four interpenetrating **dia** nets are shown in figure 20. All edges are straight, and each individual component is equivalent to an ideal embedding of a single **dia** component. Tight ropelengths produced by PB-SONO are *L*/*D*=4.81,18.00 for the two- and four-**dia**, respectively.

Our final examples are catenated patterns with two and four interpenetrating **pcu** nets (figure 21). All edges within the ideal conformation of the two-**pcu** pattern are straight and equal to the length of a cubic unit cell translation. Furthermore, both of the individual components adopt the symmetric configuration of the individual tight **pcu** net. In total, there are six distinct edges (three within each component), and the maximum possible diameter is exactly half the unit cell edge: this gives a theoretical ropelength of 12, which is exactly replicated by the simulation. In the ideal four-**pcu** pattern, the edges curve slightly. This is clearly not an artefact of our PB-SONO algorithm, as the relaxed ropelength *L*/*D*=40.46, in contrast to the corresponding value for a pattern with straight edges, *L*/*D*=48 (as there are 12 edges, each of length one, and the maximum diameter for straight edges is one-quarter).

## 7. Conclusion

This account demonstrates the potential of tightening algorithms for exploring distinct entanglements of finite graphs, as well as periodic weavings and nets—including non-crystallographic nets. Our particular implementation (PB-SONO) is shown to be accurate for the simplest entanglements. PB-SONO gives tight embeddings of periodic weavings unambiguously. However, the extension of ‘rope tightening’ algorithms to branched graphs is non-trivial. We show that PB-SONO gives reasonable configurations for relaxed polyhedral graphs.

Our analysis of tight embeddings of some crystallographic nets confirms that these embeddings are essentially equivalent to those found via the equilibrium placement (and subsequent embedding) concept [76]. The tightening procedure further allows us to define ‘canonical’ embeddings of nets with collisions and non-crystallographic nets. Likewise, the procedure allows us to describe and determine numerically tight embeddings of arrays of catenated nets in an intuitive fashion. These examples have convinced us that tight embeddings found by the PB-SONO algorithm are a useful tool for describing the entanglement of nets and embeddings of entangled graphs. There remain, however, numerical uncertainties in the current implementation of the PB-SONO algorithm, that we hope will be improved over time.

We have dodged the issue of just how ‘canonical’ these tight embeddings are. Some examples confirm that tight embeddings are not unique, though most examples of interest and relevance to atomic structure and materials (e.g. those found in the Reticular Chemistry Structure Resource [80]) do give unique tight embeddings. Indeed, such examples apparently form embeddings with the maximal symmetry consistent with their entanglement. Other examples, however, tighten to form embeddings with less than the full suite of allowed symmetries, such as the woven chainmail structure in figure 14. Furthermore, tight embeddings of some rod packings lead to curvilinear ‘rods’, in preference to the configurations with straight cylinders. We feel, however, that the very broad applicability of the tightening concept makes it a useful one in defining canonical embeddings, i.e. explicit geometries, for a range of patterns, including periodic weavings, nets and catenated nets. Moreover, it brings abstract topological objects into the physical world to inspire the study of form and function.

## Authors' contributions

M.E.E. wrote the PB-SONO code and generated the examples with advice from V.R. and S.T.H. M.E.E. generated all images. All authors contributed to the analysis and manuscript.

## Competing interests

We have no conflicts of interest.

## Funding

M.E.E. thanks the Humboldt foundation, the DFG ‘Geometry and Physics of Spatial Random Systems’ and the DFG Emmy Noether Program. V.R. thanks the Australian Research Council.

- Received April 17, 2015.
- Accepted July 29, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.