# Symmetric Tangling of Honeycomb Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**hcb**[1]. It is a motif well recognised in many disciplines across the natural sciences. In the same way that a loop can be knotted or linked in a multitude of different ways, graphs—and periodic graphs like

**hcb**—can also be entangled into a wide variety of more complicated structures. Figure 1 shows the entanglement of three

**hcb**networks in two distinct ways.

**hcb**networks. Mathematically, intrinsic topology refers to topological equivalence under homeomorphism, and extrinsic topology refers to topological equivalence via ambient isotopy. In addition to these ideas, patterns can also be described through the lens of geometry, which refers to the specific geometric arrangement. The structures in Figure 1 are symmetric, which is a geometric property. In general, the search for optimal geometric embeddings of entangled structures bridges the ideas of extrinsic topology and geometric embedding [4].

**hcb**networks have been chemically synthesised and described in various contexts [15]. The theoretical design of entangled

**hcb**networks has been explored from a few different but limited perspectives, including hyperbolic tilings [16], constructions using only straight edges [17], and other geometric techniques [18,19,20]. We will present here a comprehensive enumeration of the most symmetric entanglements of

**hcb**networks up to a given complexity, also demonstrating with an explicit construction that these entanglements can be constructed with an infinitely large number of components.

**hcb**network, and looks at different entanglements (different extrinsic topology) of that network in a systemetic way. We considered twisted helical segments of n strands arranged along the edges of an underlying graph that carries the entanglement, in this case the 2-periodic

**hcb**network, where the helical motifs are joined together in a simple way. Enumeration through the number of strands n in our helices, and their helical pitch t produces a wide variety of entangled structures with systematically increasing complexity. This technique has been used previously arranging helices on the edges of platonic polyhedra, giving an enumeration of symmetric tangled platonic polyhedra [21].

**hcb**network, but the method can be applied to any graph. This graph can be tubified, which is akin to inflating the vertices and edges of the graph like a balloon, to form a surface surrounding the original graph. Our entangled structure will sit as curves on this tubified surface. In a sense, we are utilising the symbiosis of a helical twist and a cylinder to exploit 2-dimensional surface topology for our entanglement purposes.

**hcb**within a single periodic unit cell. These helices sit as non-intersecting curves on the tubified surface. For simplicity, we dictate that all of these helices have the same number of strands; however, the technique can be applied more generally, and will be described elsewhere. The ends of these helical strands get connected around the 3-fold junction of the underlying graph: the even number of strands allows this to happen seamlessly, where exactly half of the strands will be attached to the adjacent helix on the left, and the other half of the strands to the right, without crossing over the vertical plane into the page that runs along the original graph edge. An illustration is given in Figure 2a. This process gives us that all “entanglement” is restricted to the helices, and does not occur in the connections of the strands around the junction.

**hcb**networks, provided that we enumerate through all combinations of twists and the number of strands. Those structures with the highest symmetry will come from indices of the form ${\left(\right)}^{\frac{t}{n}}$, where all twists and strand numbers are equivalent. We explore these structures here, and defer the most complete enumeration of lower symmetry cases to be published elsewhere.

**hcb**graph within this hexagon is exactly $2({a}^{2}+ab+{b}^{2})$, and these are the locations of the vertices of all other components in the structure. Given the symmetry of the structure, each component will have two vertices within the region, and we can thus conclude that the structure has exactly ${a}^{2}+ab+{b}^{2}$ components. This calculation works for prime weaves, with complete weaves having double the number of components.

## 2. Results

## 3. Discussion

**hcb**networks.

**hcb**network. In this article, these three helices are all the same, giving the high symmetry. This starting point gives several directions of increasing complexity of the structures. The first is increasing n and t in the current setting. The second is relaxing the condition that all three helices are the same: this would result in different symmetry settings, and likely cover many of the entanglements of

**hcb**observed in the synthetic chemistry literature, such as [19]. Beyond this, we can increase complexity by enumerating beyond the genus-2 structure of the tubified

**hcb**to a larger unit cell, which has a higher genus, and thus more distinct edges along which to place helices.

**hcb**structure, which has symmetry $p321$ and sits on a tubified genus-4

**hcb**surface. This structure has been derived by an anonymous referee applying the methods described in [27]. It is related to the ${\left(\right)}^{\frac{5}{12}}$ structure listed in Table 1, which has 54 components. Exactly six copies of the 9

**hcb**structure, which reticulates a genus-4 surface, will give the ${\left(\right)}^{\frac{5}{12}}$, which reticulates a genus-2 surface. We can consider this structure as one with index ${\left(\right)}^{\frac{\frac{5}{6}}{2}}$, where the larger genus is dictated by the fractional twist $\frac{5}{6}$. The structure is shown in Figure 6. Beyond this, we are also able to enumerate structure on other underlying graphs, both 2-periodic and 3-periodic. These extensions will be presented elsewhere in the near future.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Two distinct entanglements of three

**hcb**networks. The patterns have indices (

**Left**) ${\left(\right)}^{\frac{0.5}{2}}$ and (

**Right**) ${\left(\right)}^{\frac{0.5}{4}}$. The construction and characterisation of these entanglements come from the gluing of 2-fold and 4-fold helices, respectively.

**Figure 2.**(

**a**) One repeat unit of the

**hcb**network, the full network can be formed by repeating periodically. (

**b**) The tubified surface around the network is shown in pink. (

**c**) Three distinct 6-helices wind on the surface, and are connected together around the junctions of the surface, where three strands pass to the right of the helix and three to the left (as you travel along the underlying graph edge. (

**d**) The blue network is constructed as the medial axis of the region on the surface defined by the black curves.

**Figure 3.**(

**Left**) The curved lines of the ${\left(\right)}^{\frac{1}{3}}$ structure as they wind around the tubified surface of the underlying

**hcb**network. (

**Right**) When this motif is repeated in a periodic way, the global arrangement is the entanglement of seven

**hcb**networks, each with a different colour. The pattern has in-surface orbifold 2233, 2D planar symmetry 236 and layer group $p6$.

**Figure 4.**A single tube of the surface along which our strands wind, we can see that twisting an amount that is below $\frac{n}{2}$ means that the strand will exit to the right, and when it twists beyond $\frac{n}{2}$ (which is then symbolised as a negative number in the string), it will then exit to the left. Hence positive numbers in our string symbolise a right handed turn, and negative numbers left handed turns.

**Figure 5.**(

**Left**) The path of left and right handed turns that an edge of the ${\left(\right)}^{\frac{3.5}{9}}$ structure takes through the underlying tiling. The resulting Goldberg vector is $(7,0)$, which is obtained by the traversing of two edges. (

**Centre**) The width of a single hexagon in the underlying graph is set to 1, and this hexagon has an area of $\frac{3\sqrt{3}}{2}$. (

**Right**) The Goldberg vector defines the width of the larger hexagon defined by a single component of the entangled structure. The length of this vector, c, given coordinates of $(a,b)$ is ${c}^{2}={a}^{2}+ab+{b}^{2}$, which means that the area of this larger hexagon is $\frac{3\sqrt{3}}{2}{c}^{2}=\frac{3\sqrt{3}}{2}({a}^{2}+ab+{b}^{2})$.

**Figure 6.**(

**Left**) A nine-component

**hcb**structure, which sits on a tubified genus-4

**hcb**surface. This surface has nine distinct edges along which helices lie, which then branch around 6 vertices. This structure will have an index of ${\left(\right)}^{\frac{\frac{5}{6}}{2}}$. This structure has symmetry $p321$. (

**Right**) The extended structure, showing the nine

**hcb**components.

**Table 1.**Tabulation of the results of the constructive technique, ordered by the Goldberg vectors of the entangled structures, showing the number of components. The first row associated with each vector refers to the prime structures, and the second row the complete structures.

Goldberg | Components | Layer Group | Examples (By Twist Indices, Where $\mathit{a}\in \mathbb{Z}$, $\mathit{b}=1,2,\dots $) |
---|---|---|---|

(1,0) | 1 | $p6$ or $p321$ | ${\left(\right)}^{\frac{0}{1}}$, ${\left(\right)}^{\frac{5a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.5}{5}}$, ${\left(\right)}^{\frac{7a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.5}{7}}$, ${\left(\right)}^{\frac{(6b\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1)a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.5}{(6b\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1)}}$ |

2 | $p622$ | ${\left(\right)}^{\frac{a}{2}}$ | |

(1,1) | 3 | $p312$ | ${\left(\right)}^{\frac{a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.5}{2}}$, ${\left(\right)}^{\frac{4a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.5}{4}}$, ${\left(\right)}^{\frac{6a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1.5}{6}}$, ${\left(\right)}^{\frac{8a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.5}{8}}$ |

6 | $p622$ | ${\left(\right)}^{\frac{4a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1}{4}}$, ${\left(\right)}^{\frac{8a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1}{8}}$ | |

(2,0) | 4 | $p321$ | ${\left(\right)}^{\frac{3a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.5}{3}}$, ${\left(\right)}^{\frac{9a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.5}{9}}$, ${\left(\right)}^{\frac{9a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1.5}{9}}$ |

8 | $p622$ | ${\left(\right)}^{\frac{6a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1}{6}}$, ${\left(\right)}^{\frac{18a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1}{18}}$, ${\left(\right)}^{\frac{18a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3}{18}}$ | |

(2,1) | 7 | $p6$ | ${\left(\right)}^{\frac{3a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1}{3}}$, ${\left(\right)}^{\frac{9a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1}{9}}$ |

14 | $p622$ | ${\left(\right)}^{\frac{6a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2}{6}}$, ${\left(\right)}^{\frac{18a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2}{18}}$ | |

(2,2) | 12 | $p312$ | ${\left(\right)}^{\frac{4a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1.5}{4}}$, ${\left(\right)}^{\frac{12a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}4.5}{12}}$ |

24 | $p622$ | ${\left(\right)}^{\frac{8a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3}{8}}$, ${\left(\right)}^{\frac{24a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}9}{24}}$ | |

(3,0) | 9 | - | |

18 | - | ||

(3,1) | 13 | $p6$ | ${\left(\right)}^{\frac{5a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1}{5}}$, ${\left(\right)}^{\frac{7a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1}{7}}$, ${\left(\right)}^{\frac{(6b\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1)a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1}{(6b\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1)}}$ |

26 | $p622$ | ${\left(\right)}^{\frac{(6b\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1)2a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2}{2(6b\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1)}}$ | |

(3,2) | 19 | $p6$ | ${\left(\right)}^{\frac{5a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2}{5}}$ |

38 | $p622$ | ${\left(\right)}^{\frac{10a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}4}{10}}$ | |

(3,3) | 27 | $p312$ | ${\left(\right)}^{\frac{6a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2.5}{6}}$, ${\left(\right)}^{\frac{8a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1.5}{8}}$, ${\left(\right)}^{\frac{8a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2.5}{8}}$, ${\left(\right)}^{\frac{10a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1.5}{10}}$, ${\left(\right)}^{\frac{10a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3.5}{10}}$ |

54 | $p622$ | ${\left(\right)}^{\frac{12a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}5}{12}}$, ${\left(\right)}^{\frac{16a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3}{16}}$, ${\left(\right)}^{\frac{16a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}5}{16}}$, ${\left(\right)}^{\frac{20a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3}{20}}$, ${\left(\right)}^{\frac{20a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}7}{20}}$ | |

(4,0) | 16 | $p321$ | ${\left(\right)}^{\frac{5a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1.5}{5}}$ |

32 | $p622$ | ${\left(\right)}^{\frac{10a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3}{10}}$ | |

(4,1) | 21 | - | |

42 | - | ||

(4,2) | 28 | - | |

56 | - | ||

(4,3) | 37 | $p6$ | ${\left(\right)}^{\frac{7a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3}{7}}$ |

74 | $p622$ | ${\left(\right)}^{\frac{14a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}6}{14}}$ | |

(4,4) | 48 | $p312$ | ${\left(\right)}^{\frac{8a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3.5}{8}}$, ${\left(\right)}^{\frac{12a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2.5}{12}}$ |

96 | $p622$ | ${\left(\right)}^{\frac{16a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}7}{16}}$, ${\left(\right)}^{\frac{24a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}5}{24}}$ |

**Table 2.**Tangled

**hcb**weavings, ordered by the Goldberg vectors of the entangled structures, and showing the number of components. The first row associated with each vector refers to the prime structures, and the second row the complete structures.

Goldberg | Components | Layer Group | Examples (By Twist Indices, Where $\mathit{a}\in \mathbb{Z}$) |
---|---|---|---|

(5,0) | 25 | $p321$ | ${\left(\right)}^{\frac{7a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1.5}{7}}$, ${\left(\right)}^{\frac{7a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2.5}{7}}$, ${\left(\right)}^{\frac{11a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1.5}{11}}$ |

50 | $p622$ | ${\left(\right)}^{\frac{14a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3}{14}}$, ${\left(\right)}^{\frac{14a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}5}{14}}$, ${\left(\right)}^{\frac{22a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3}{22}}$ | |

(5,1) | 31 | $p6$ | ${\left(\right)}^{\frac{7a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2}{7}}$ |

62 | $p622$ | ${\left(\right)}^{\frac{14a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}4}{14}}$ | |

(5,3) | 49 | $p6$ | ${\left(\right)}^{\frac{11a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2}{11}}$ |

98 | $p622$ | ${\left(\right)}^{\frac{22a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}4}{22}}$ | |

(5,4) | 61 | $p6$ | ${\left(\right)}^{\frac{9a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}4}{9}}$ |

122 | $p622$ | ${\left(\right)}^{\frac{18a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}8}{18}}$ | |

(5,5) | 75 | $p312$ | ${\left(\right)}^{\frac{10a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}4.5}{10}}$, ${\left(\right)}^{\frac{12a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3.5}{12}}$ |

150 | $p622$ | ${\left(\right)}^{\frac{20a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}9}{20}}$, ${\left(\right)}^{\frac{24a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}7}{24}}$ | |

(6,0) | 36 | - | |

72 | - | ||

(6,1) | 43 | $p6$ | ${\left(\right)}^{\frac{9a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2}{9}}$ |

86 | $p622$ | ${\left(\right)}^{\frac{18a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}4}{18}}$ | |

(7,0) | 49 | $p321$ | ${\left(\right)}^{\frac{9a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}2.5}{9}}$, ${\left(\right)}^{\frac{9a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}3.5}{9}}$ |

98 | $p622$ | ${\left(\right)}^{\frac{18a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}5}{18}}$, ${\left(\right)}^{\frac{18a\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}7}{18}}$ |

**Table 3.**Selected patterns from Table 1, listed by constituent helices $\frac{t}{n}$, the number of distinct nets, and the corresponding layer group symmetry. Crystallographic files (CIF) are given for each of these structures as supplementary information.

${\left(\right)}^{\frac{0}{1}}3$ − 1 × hcb | ${\left(\right)}^{\frac{0.5}{5}}3$ − 1 × hcb − $p321$ | ${\left(\right)}^{\frac{1}{2}}3$ − 2 × hcb − $p622$ |

${\left(\right)}^{\frac{0.5}{2}}$ − 3 × hcb − $p312$ | ${\left(\right)}^{\frac{0.5}{3}}$ − 4 × hcb − $p321$ | ${\left(\right)}^{\frac{1}{4}}$ − 6 × hcb − $p622$ |

${\left(\right)}^{\frac{1}{3}}$ − 7 × hcb − $p6$ | ${\left(\right)}^{\frac{1}{6}}$ − 8 × hcb − $p622$ | |

**Table 4.**Selected patterns from Table 1. Crystallographic files (CIF) are given for each of these structures as supplementary information.

${\left(\right)}^{\frac{1.5}{4}}3$ − 12 × hcb − $p312$ | ${\left(\right)}^{\frac{1}{5}}3$ − 13 × hcb − $p6$ |

${\left(\right)}^{\frac{2}{6}}$ − 14 × hcb − $p622$ | ${\left(\right)}^{\frac{1.5}{5}}$ − 16 × hcb − $p321$ |

**Table 5.**Two structures with 49 components, but different Goldberg vectors. It could be assumed that the distinct Goldberg vectors imply that the structures are not ambient isotopic to each other. The upper two images show a single component highlighted in yellow, and the bottom two images show each of the 49 nets coloured differently. Crystallographic files (CIF) are given for each of these structures as supplementary information.

${\left(\right)}^{\frac{2}{11}}3$ − 49 × hcb − $p6$ | ${\left(\right)}^{\frac{2.5}{9}}3$ − 49 × hcb − $p321$ |

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Evans, M.E.; Hyde, S.T.
Symmetric Tangling of Honeycomb Networks. *Symmetry* **2022**, *14*, 1805.
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Evans ME, Hyde ST.
Symmetric Tangling of Honeycomb Networks. *Symmetry*. 2022; 14(9):1805.
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Evans, Myfanwy E., and Stephen T. Hyde.
2022. "Symmetric Tangling of Honeycomb Networks" *Symmetry* 14, no. 9: 1805.
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