# The Symmetry and Topology of Finite and Periodic Graphs and Their Embeddings in Three-Dimensional Euclidean Space

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{6}tilings of the cylinder. The simplest of these is the Boerdijk–Coxeter helix, which has vertices related by screw rotations of cos

^{−1}(−3/2), which is not a rational fraction of a circle [3]. The duals of these structures are the structures of chiral carbon nanotubes, surely also lacking translational periodicity. Other excluded structures are quasicrystals [4], structures with two or more incommensurate periodicities [5], and non-rigid objects [6]. We are instead particularly concerned here to make the case (a) for using the ‘international’ system (Hermann–Mauguin notation) to cover systematically all dimensions and periodicities within three-dimensional Euclidean space: and (b) clearly distinguishing abstract graphs from their realizations as embeddings in Euclidean space.

## 2. The Space Groups and Subperiodic Groups

$N\mathrm{odd}:$ | $pN$ | $p{N}_{i}$ | $p\overline{N}$ | $pN2$ | $p{N}_{i}2$ | $pNm$ | $pNc$ | $p\overline{N}m$ | $p\overline{N}c$ |

$N\mathrm{even}:$ | $pN$ | $p{N}_{i}$ | $p\overline{N}$ | $pN/m$ | $p{N}_{k}/m$ | $pN22$ | $p{N}_{i}22$ | $pNmm$ | $p{N}_{k}cm$ |

$pNcc$ | $p\overline{N}2m$ | $p\overline{N}2c$ | $pN/mmm$ | $pN/mcc$ | $p{N}_{k}/mmc$ |

**c**.

**a**,

**b**plane with possible lattices p and c, and only those groups with no translations along

**c**, such as screw and glide operations, are accepted. The layer groups have been known for over 100 years, for references see [12], and are listed in terms of HM notation in a number of places, e.g., [9,10,11,12]. We emphasize this last point because, although 2-periodic objects are common in everyday experience (weaving, knitting, chain mail, chain-link fencing) and have occasioned mathematical discussions, we have not found any work prior to our recent work (discussed below) that actually describes their symmetries correctly. A notable exception is an enumeration, and crystallographic description, of bilayers of vertex-transitive sphere packings [13].

## 3. Symmetries of Embedded Finite Graphs—Knots, Links, and Tangles: Terminology

_{2}O) and heavy water (D

_{2}O) are isotopic molecules as they contain isotopes (atoms of the same atomic number, but different mass) of the same element. Anisotopic is also found to refer to an element with only one stable isotope, e.g., fluorine.

_{3,3}. This graph contains two sets of three vertices, A and B, with edges linking all A to all B, but no A–A or B–B edges occur. This is also known as the 3-Möbius ladder graph and has occasioned considerable discussion by chemists [25]. The order of the automorphism group is readily found as follows: permuting the A vertices (3! permutations) or the B vertices (another 3!) does not change the adjacencies. Nor does interchanging A and B. The total number of graph-preserving permutations is then 3! × 3! × 2 = 72. An embedding in three-dimensional Euclidean space with symmetry of that order is impossible. The rotational symmetry would have to be at least 18-fold (in 18/mmm), which is impossible with only 6 vertices. The highest-symmetry embeddings of the graph we can find have symmetry $32$ (order 6, three crossings) and $\overline{4}$ (order 4, one crossing) as shown in Figure 3.

_{,}the cyclic groups of order $n$; ${D}_{n}$, the dihedral groups of order $2n$; ${S}_{n}$, the group of $n$ permutations of order $n!$, and ${A}_{n}$, the alternating group of even permutations of order $n!/2$. Note that ${S}_{3}$ is the same as ${D}_{3}$, and ${A}_{3}$ is the same as ${Z}_{3}$. The structure of the cubic and icosahedral groups in terms of these abstract groups are:

## 4. Symmetries of 1-Periodic Embedded Graphs and Weavings

^{6}tilings, by triangles and their duals, 6

^{3}tilings by hexagons. The vertices of 3

^{6}tilings are the centers of cylindrical close packings of equal spheres, and a detailed account of the geometry of such structures, and their occurrence in biological structure, was given almost 50 years ago [31]. More recently, the dual structures, 6

^{3}tilings, have attracted considerable attention as the structures of carbon tubules (“nanotubes”) [32].

^{6}tilings. The tiling (graph) can be specified by its derivation from the plane tiling. For the plane tiling, the hexagonal cell has axes

**a**and

**b**at 120°. The tilings are specified by the girdle vector $\left[u,v\right]=ua+vb$, which is orthogonal to, and goes around, the cylinder axis. They fall into two groups. Achiral structures, [u, 0] and [u, 2u], have maximum symmetry $p{(2N)}_{N}/mcm$ with $2N$ vertices in the unit cell. Chiral tilings by regular triangles (sphere packings with six contacts) are not periodic, but the question of the possible symmetries of the hexavalent graph is of interest. We note that, in the carbon nanotube literature, it is common to use instead axes at 60°, and specify tilings as $\left(m,n\right)$ with $m=u-v$, $n=v$.

^{−1}(−2/3) $\approx $ 131.8°. In a $p{8}_{3}22$ embedding, the rotation along the axis is 360 $\times $ 3/8 = 135° so, with that structure, approximately equal edges are possible, and indeed that embedding in the subgroup $p{4}_{3}22$ is found in structures such as β-Mn [3], and metal-organic frameworks [30].

^{6}tilings given above are all isotopic—one can be transformed into another by a simple twist. In Figure 7 we show some different embeddings of the [3,0] 6

^{3}nanotube. These, in contrast, are anisotopic. In one case, a 6-ring is replaced by a 3

_{1}(trefoil) knot, and in another 6-rings are catenated.

^{4}tilings with symmetry $p{(2N)}_{N}/mcm$ ($N>1$); the case $N=3$ is shown in Figure 7.

## 5. Symmetries of 2-Periodic Graphs and Weavings

**c**and that, when the lower-case letter “c” appears at the beginning of a group symbol, it refers to the c-centered lattice. The crystallographic constraint applies, so there are only 80 layer groups. The symbols are the same as those for the corresponding 3-periodic groups, except that the lattice symbol is lower case. In this section, we show isogonal piecewise linear embeddings of 2-periodic patterns taken from recent work [15,35].

**fxt**, one of just 4 such nets (Figure 10). It is the net of a tiling of the plane by squares, hexagons, and dodecagons, vertex symbol 4.6.12. If the hexagons were replaced by trefoil knots, we obtain a vertex-transitive anisotopic tangled embedding of the graph

**fxt-z**(Figure 10, center). The dodecagon can also be replaced by a knot; the only vertex-transitive possibility is the 24-crossing torus knot T(6,5). This gives the embedding labeled

**fxt-z***in Figure 10. Interestingly, in this embedding, the hexagons alone are linked into a chain mail pattern. We emphasize that these are three topologically distinct (ambient anisotopic) embeddings of the same graph.

## 6. Symmetries of 3-Periodic Graphs and Their Embeddings

**xyz**, sometimes with an extension as in

**xyz-q**. It is very common to see, in the chemical literature, statements such as “structures with the

**pts**topology” [45], and the RCSR symbols have to be interpreted as referring both to a graph (net) and also to a particular embedding of that graph (a “topology”). In RCSR, crystallographic nets (graphs) are normally given an embedding (topology) with full symmetry. However, sometimes an alternative embedding is given and identified by an appended

**-z**. An example is the 5-valent net

**gan**with symmetry $Ia\overline{3}d$ [36]. However, in that embedding, all edges cannot have equal length and the five-coordinated sphere packing 5/5/c1, which has the same net and equal edges, has symmetry $I\overline{4}3d$. RCSR assigns that embedding the symbol

**gan-z**. So far, no problem, but the ubiquitous

**dia**net (the net of the diamond structure) with symmetry $Fd\overline{3}m$, also occurs in a tangled (not isotopic) version with symmetry $R\overline{3}m$ [46]. The shortest cycles in the

**dia**net are 6-rings, and in the

**dia**embedding, they are not catenated with each other as shown in Figure 11. However, as also shown in Figure 11, in

**dia-z**the 6-rings are catenated, so that embedding is topologically distinct (anisotopic) to the

**dia**embedding. We emphasize that programs such as Systre and ToposPro identify the graph, not the topology of the embedding. ToposPro can often distinguish between two topologically-distinct embeddings of the same graph, such as

**dia**and

**dia-z**, based on the catenation of rings. However, not all tangles contain catenated rings.

**pcu**, of the primitive cubic lattice with symmetry $Pm\overline{3}m$, has no catenated 4-rings in its maximum-symmetry embedding, but there is a topologically distinct embedding,

**pcu-z**, with symmetry $Pa\overline{3}$ in which the 4-rings are catenated. Another commonly occurring net is the trivalent

**srs**(symmetry $I{4}_{1}32$), which contains cages of three 10-rings. In the embedding

**srs-z**(symmetry P4

_{1}32), the cage graph is entangled, as shown in Figure 12. Interestingly, both these tangled embeddings are vertex- and edge-transitive.

**dia-c**for a pair of interpenetrating diamond nets., and

**dia-c3**,

**dia-c3***, and

**dia-c3****for three distinct anisotopic structures comprising three distinct interpenetrating diamond nets.

## 7. Nets with Edge-Intersecting, or Vertex-Colliding, Embeddings

_{2}provides a good illustration [38]. Figure 13 shows the structure of the real material, symmetry C2/c. and the pattern of anions as a net of corner-connected tetrahedra. In the maximum-symmetry embedding, Cmmm, some tetrahedra become planar rectangles, and to accommodate tetrahedral shape one must go to a doubled cell with symmetry Ibam (an order two subgroup of Cmmm) as shown in Figure 13. Similar phenomena occur in the anion nets of some zeolite nets [54].

**cya**illustrated in Figure 14. This is one of a group of such nets of importance in reticular chemistry [55]. Clearly, in this case, there are non-rigid-body symmetries (interchanging pairs of vertices) and Systre does not handle such cases. A slightly more complicated case is the minimal net labeled 4(3)4 [56]. In this case, in barycentric coordinates, groups of four vertices collide and edges shrink to zero length, [57].

**mhq**is shown in maximum symmetry without collisions: $Fm\overline{3}m$. In barycentric coordinates groups of four vertices collide, and the symmetry is now $Pm\overline{3}m$, with half the unit cell edge. The second example here is a remarkable net

**rld**. In the reported structures it has symmetry $Pbcn$ with vertices (eight per cell) falling on four interpenetrating diamond (

**dia**) nets linked by an extra edge to make a five-coordinated structure (Figure 15). Upon relaxation of the net to barycentric coordinates, that fifth edge shrinks to zero length, and the symmetry becomes $Im\overline{3}m$ with just two vertices in the primitive cell.

**qld**shown in an embedding with symmetry $I432$. In barycentric coordinates, the vertices come together in pairs, and edges coincide. The full symmetry is $I432$ combined with the “ladder” operation of interchanging all pairs of vertices. The second example in Figure 16,

**uld-z**, is the net of a vertex-transitive sphere packing with symmetry $I{4}_{1}32$. The net contains “ladders” shown with red and blue “risers” and green “rungs”. In barycentric coordinates, the rungs shrink to zero length and the ladder “risers” collide. The symmetry is $I{4}_{1}32$, combined with the “ladder” automorphism of interchanging risers. Systre can derive a key for ladder nets so such nets (graphs) can be unambiguously identified.

## 8. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Embeddings of the cube graph and their symmetries.

**Top row**: ambient isotopes, all without essential crossings.

**Bottom row**: topologically-distinct embeddings of the cube graph—anisotopes with different numbers of crossings. Blue edges outline a substructure that is a knot or link.

**Figure 3.**Two embeddings of the bipartite graph K

_{3,3}. Edges join vertices of opposite color only.

**Figure 4.**

**Top row**: topologically-distinct embeddings of the Franklin graph showing the three 4-rings as red, blue, and green.

**Bottom row**: the same with numbered vertices.

**Figure 5.**Possible unit cells for the (3,1) 3

^{6}cylinder tiling. The heavy black line is the [3,1] vector (using a 120° interaxial angle). The unit cell at the bottom is rectangular.

**Figure 6.**

**Top**: Various embeddings of the (3,1) 3

^{6}cylinder tiling. Vertical magenta lines indicate unit cell edges.

**Bottom left**: the (3,0), and

**right**, the (6,3) tilings.

**Figure 7.**Embeddings of the hexagonal zigzag 6

^{3}cylinder tiling. From top: untangled; interpenetrating pair; tangled with knots; tangled with links (e.g., red and blue). The bottom is a hexagonal, vertex- and edge-transitive, 4

^{4}tiling.

**Figure 11.**(

**a**) A fragment of the

**dia**embedding of the diamond graph, and (

**b**) a corresponding fragment of

**dia-z**. (

**c**) The pattern (adamantane cage) of 6-rings in

**dia**, and (

**d**) the corresponding pattern in

**dia-z**. (

**e**) Showing two catenated rings in

**dia-z**.

**Figure 12.**

**Left**: fragments of the

**srs**and

**pcu**nets in their maximum-symmetry embeddings.

**Center**and

**right**: fragments of a tangled version of these nets.

**Figure 13.**

**Bottom left**: the crystal structure of moganite, SiO

_{2}(silicon cations are blue; oxygen anions are red).

**Top left**: the 6-valent anion net (Si atoms at the tetrahedra centers; O atoms at the tetrahedra apices).

**Top right**: the maximum-symmetry embedding of the anion net—$Cmmm$; some “tetrahedra” are planar, a configuration that does not arise in silicates.

**Bottom right**: the maximum-symmetry anion net re-embedded as regular tetrahedra. The two shades of blue indicate the crystallographically-distinct Si atoms. The cell height is doubled, and the symmetry group is now $Ibam$, an order-2 subgroup of $Cmmm$.

**Figure 14.**Nets with next-neighbor collisions. In

**cya**pairs collide. In 4(3)4 groups of four collide, and there are zero-length bonds.

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O’Keeffe, M.; Treacy, M.M.J.
The Symmetry and Topology of Finite and Periodic Graphs and Their Embeddings in Three-Dimensional Euclidean Space. *Symmetry* **2022**, *14*, 822.
https://doi.org/10.3390/sym14040822

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O’Keeffe M, Treacy MMJ.
The Symmetry and Topology of Finite and Periodic Graphs and Their Embeddings in Three-Dimensional Euclidean Space. *Symmetry*. 2022; 14(4):822.
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https://doi.org/10.3390/sym14040822