#
Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

_{C}of these nano-structures, which have a constant length L

_{M}of the Möbius zigzag edge. The presented results show that Klein bottle cubic graphs are topologically indistinguishable from toroidal lattices with the same size (N, L

_{C}, L

_{M}) over a certain threshold size L

_{C}. Both nano-structures share the same values of the topological indices that measure graph compactness and roundness, two key topological properties that largely influence lattice stability. Moreover, this newly conjectured topological similarity between the two kinds of graphs transfers the translation invariance typical of the graphenic tori to the Klein bottle polyhexes with size L

_{C}≥ L

_{C}, making these graphs vertex transitive. This means that a traveler jumping on the nodes of these Klein bottle fullerenes is no longer able to distinguish among them by only measuring the chemical distances. This size-induced symmetry transition for Klein bottle cubic graphs represents a relevant topological effect influencing the electronic properties and the theoretical chemical stability of these two families of graphenic nano-systems. The present finding, nonetheless, provides an original argument, with potential future applications, that physical unification theory is possible, starting surprisingly from the nano-chemical topological graphenic space; thus, speculative hypotheses may be drawn, particularly relating to the computational topological unification (that is, complexification) of the quantum many-worlds picture (according to Everett’s theory) with the space-curvature sphericity/roundness of general relativity, as is also currently advocated by Wolfram’s language unification of matter-physical phenomenology.

## 1. Introduction

## 2. Method: Topological Invariants for Polyhex Graphs

_{M}and L

_{C}indicate, respectively, the lengths of the Möbius-like edge (subject to the antiparallel sewing) and the cylindrical edge (parallel sewing), both expressed as the number of hexagons. The graphenic open lattice ${\mathrm{G}}_{{L}_{M},{L}_{C}}$ is made with ${L}_{C}$ rows with ${L}_{M}$ hexagons. The number h of total hexagons in the ${L}_{C}$ belts is then $h={L}_{M}\times {L}_{C}$. By closing the edges, the numbers of atoms N and bonds B in the graph are: N $=4h$,$B=\frac{3\mathrm{N}}{2}=6h$.

_{M}=3, L

_{C}=1 honeycomb network. This graphenic lattice may be closed in two distinct topological manners, forming the torus T

_{3,1}or the Klein bottle KB

_{3,1}where $h=3,N=12,\text{}\mathrm{and}\text{}B=18$. In both cases, the L

_{C}hexagons are closed in parallel along x. On the other edge, the dangling bonds belonging to the L

_{M}hexagons are closed in parallel or antiparallel along y to form the T

_{3,1}torus or the KB

_{3,1}Klein bottle. Figure 3 shows more detail of how the two closed structures T

_{3,1}and KB

_{3,1}are built. The populations over the coordination shells change in a significant way depending on which of the two structures is considered.

_{3,1}have the same set {${b}_{ik}$} = {3,4,3,1}, with one node j in the fourth coordination shell at the maximum distance d

_{ij}= 4. The maximum distance from i determines the node eccentricity ${\epsilon}_{i}=4$. The Klein bottle KB

_{3,1}shows some “toroidal” nodes with the same {${b}_{ik}$} set (namely the vertices 2, 3, 5, and 8), and the remaining nodes have a reduced eccentricity ${\epsilon}_{i}=3$ and {${b}_{ik}$} = {3,5,3}. For example, vertex 1 in Figure 2 has three neighbors—4, 8, and 10—in the third coordination shell but zero nodes in the fourth shell.

_{i}is defined in such a way:

_{60}or C

_{84}[21,22] tend to minimize the ${\rho}_{E}$ topological descriptor. Tori ${\mathrm{T}}_{{L}_{M},{L}_{C}}$ have ${\rho}_{E}=1$, which is different from the situation for the Klein bottles. In our example, ${\rho}_{E}\left(K{B}_{3,1}\right)=\frac{12}{11}>{\rho}_{E}\left({T}_{3,1}\right)$, evidencing the greater topological stability of the torus.

_{.}This is an evidently arbitrary choice intended only to represent all cases with ${L}_{C}<{L}_{M}-1$.

_{6,2}possesses an augmented compactness over T

_{6,2}, that is, W(KB) < W(T). It is worth noting that W(T) $=N\overline{w}$. The Klein bottle fullerene $K{B}_{6,2}$ has a broader range of topological transmission values, from w = 91 to $\overline{w}=98$. Table 1 details the list of topological descriptors for each node of the $K{B}_{6,2}$ graph, including eccentricity and {${b}_{ik}$} sets; descriptors for ${T}_{6,2}$ are also provided for completeness (vertices are labeled as shown in Figure 3). Topological descriptors group the N = 48 nodes of the ${\mathrm{KB}}_{6,2}$ graph into four classes of topological equivalent vertices, with the multiplicity given in brackets in the first column of Table 1. The

^{13}C NMR resonance spectrum of the carbon C

_{48}nanostructure has such a Klein bottle topology made of four peaks with relative intensities 1, 1, 2, and 2. We end these considerations with Table 2, which shows the eccentricity shrinkage in passing from ${T}_{3l,l}$ to $K{B}_{3l,l}$ graphs for l = 1, 2,…, 9, 10.

**Polyhex similarity conjecture.**For a given integer value L

_{M}, the shrinkage of the topological eccentricity does not hold over the threshold L

_{C}≥ Ł

_{C}with Ł

_{C}= L

_{M}− 1.

_{C}≥ Ł

_{C}

_{C}= L

_{M}− 1

_{C}that makes toroidal and Klein bottle polyhexes topologically indistinguishable and vertex transitive.

## 3. Results: Topological Similarities between Toroidal and Klein Bottle Polyhexes

_{3,2}, that is, a graph with L

_{M}= 3, and L

_{C}= 2 with Ł

_{C}= 2, fully compatible with the conditions of Equation (5), leading to the topological similarity with the torus T

_{3,2}. We start with the graphenic open lattice ${\mathrm{G}}_{3,2}$ made by N = 24 vertices, and we first close the y edge in the usual cylindrical manner, with, for example, node 14 bonding vertices 13, 15, and 21, and so on. We know from the Klein bottle KB

_{3,1}represented in Figure 2 that some nodes (such as vertex 4) show $\underset{\_}{\epsilon}$ $=3$ shrunk eccentricity with respect to the node of the torus T

_{3,1}, having eccentricity $\overline{\epsilon}=4$. In particular, vertex 4 in the Klein bottle has three vertices in the maximum distance coordination shell d

_{4,1}= d

_{4,5}= d

_{4,11}= 3. By contrast, in the torus T

_{3,1}, the same vertex still has three nodes in the third coordination shell d

_{4,9}= d

_{4,5}= d

_{4,11}= 3 but also one node 10 at the maximum distance d

_{4,10}= 4 = $\overline{\epsilon}$. When the graphenic open lattice ${G}_{{L}_{M},{L}_{C}\ge {\u0141}_{C}}$ is built respecting the conditions of Equations (5b,5c) relating to the sizes of the two edges, the topological distances of the toroidal and Klein bottle polyhexes will change in such a way that all the nodes of both graphs show the same eccentricity $\overline{\underset{\_}{\epsilon}}$ = $\underset{\xaf}{\epsilon}$ and the same set of coordination numbers {b

_{k}} with k = 1,2,..,$\underset{\_}{\epsilon}$. Klein bottle polyhexes $K{B}_{{L}_{M},{L}_{C}\ge {\u0141}_{C}}$ are made by the vertex transitive graph with the same invariants (1,2,3) shown by the nodes of the ${T}_{{L}_{M},{L}_{C}\ge {\u0141}_{C}}$ tori.

_{3,2}torus and KB

_{3,2}graphs in Figure 4. For the torus, the three nodes with distances equal to 4 are vertices 10, 16, 20, 22, and 24, and one node is at distance d

_{4,23}= 5 = $\overline{\epsilon}$. In the Klein bottle KB

_{3,2}, the antiparallel closure makes node 23 closer to node 4, at d

_{4,23}= 3, with four nodes in the fourth coordination shell—10, 16, 18, 20, and 24—and one node at the maximum distance d

_{4,19}= 5. This example fortifies the conjectured similarity among tori and KBs when L

_{C}≥ Ł

_{C}.

_{C}= L

_{M}− 1 of Equation (5c) making toroidal and Klein bottle polyhexes topologically indistinguishable. For the two families of graphs ${\mathrm{KB}}_{{L}_{M},{L}_{C}}$, L

_{M}= 5,6 and L

_{C}= 1, 2, …; the eccentricity {${\u03f5}_{i}$} values are compared with the eccentricity $\overline{\epsilon}$ of the isomeric tori. In both cases, the topological shrinkage of the eccentricity $\underset{\_}{\epsilon}<\overline{\epsilon}$ takes place in the region L

_{C}< Ł

_{C}, confirming the outcome of the current study. We also note that $\overline{\epsilon}$ = $\underset{\_}{\epsilon}$ = 2L

_{C}for L

_{C}> Ł

_{C}in both cases when L

_{M}= 5,6.

_{C}, L

_{M}) when L

_{C}≥ Ł

_{C}; see Equation (5c).

_{k}} that characterize the vertices of the isomeric torus ${T}_{{L}_{M},{L}_{C}}$; (ii) that the Klein bottle graph has become vertex transitive with all nodes sharing the same {b

_{k}}.

- ➢
- L
_{C}< Ł_{C}: in this region, the eccentricity shrinkage of Equation (4b) holds and the Klein bottle is therefore more compact than the toroidal graphenic structure with equal size. The Wiener indices (2) of the Klein bottles are lower than those of the tori; see Table 3. - ➢
- L
_{C}≥ Ł_{C}: in this region, the conjectured topological similarity of Equation (5a) holds and the Klein bottle thus becomes topologically equivalent to the isomeric toroidal polyhex sharing the same values of W and ${\rho}_{E}$; see Figure 5.

_{C}≥ Ł

_{C}, the same values of graph compactness and roundness (i.e., similarity conjecture), which are two key topological properties that largely influence the stability of the systems [21,22].

_{C}= L

_{M}− 1 of Equation (5c), the toroidal and Klein bottle polyhexes are topologically indistinguishable. Thus, they coexist, being superimposed or topologically entangled, in terms of quantum information theory. Accordingly, the two graphenic nano-systems may be quantum interrelated by the inter-correlated (Klein bottle, KB, and tori, T) wave functions, for example, by the entangled wave functions’ symmetry transition:

_{KB}({x})Ψ

_{T}({y − λ

_{T}E

_{KB}t

_{+}})dτ = ∫Ψ

_{KB}({x − λ

_{KB}E

_{T}t_})Ψ

_{T}({y})dτ

_{KB}and E

_{T}as the forward and reversal times of t

_{+}and t_ flows, and with the respective ordering (i.e., observably related) of parameters of λ

_{T}and λ

_{KB}. The future challenge will be to properly assess the topological parameters of the manifold observable many-world physical factors (e.g., establishing the degree to which the nano-structural and topological critical extended parameter L

_{C}accounts for the forward and reversal time, in a topo-quantum space–time unification approach, or the degree to which the topological roundness ρ

_{E}influences the observable (e.g., thermochemical, nuclear, magnetic, and electronic) parameters through the order parameters λ

_{KB,T}, and so on).

## 4. Conclusions

_{C}≥ Ł

_{C}is imposed on the systems. In particular, under this condition, the translation invariance typical of the graphenic tori also becomes a topological property of the Klein bottle polyhexes, thus making these Klein bottle graphs vertex transitive, and suggesting that they may have the same topological stability of graphene. Therefore, they may be synthesizable as actual chemical structures. This is an intriguing possibility that is worthy of further theoretical and experimental (quantum) investigation.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. On Topological and Quantum Coverings of Nano-Space

**Figure A1.**Upper: The (6,3) covering the toroidal and cylindrical embedding, respectively. (

**A**) (6,3)H/Z [12,50]; N = 600; (

**B**) Tu(6,3)A[12,12]; N = 144 [49]. Bottom: The Klein bottle 3D embedding representations. (

**C**) The so called “Figure 8 immersion”; (

**D**) The “Möbius strip(s) cutting(s)” contained in the Klein bottle’s gluing diagram—allowing its 3D representation by gluing both pairs of opposite edges, giving one pair a half-twist [2]; see the text for details.

**i**+ c

**j**+ d

**k**with the following coordinates: the inside-out position (by “a”, e.g., a = 0,1, via quantum logic and, thus, the quantum information approach); position b∈[0,I] along the length (I) of the Klein bottle diagram; position c∈[0,J] along the height (J) of the Klein bottle diagram; and “the 4D completeness information” d associated with the in-point-assignment (e.g., by 4D rotation algebra space or by Pauli matrices’ spin algebra space) of any information contained therein, associated with the gluing diagram. Once the algebraic assignment is completed, one can pursue a finite-dimensional quaternionic quantum formalism for the description of quantum states, quantum channels, and quantum measurements [61,62,63,64,65,66]. In addition, the present results show that the next quantum frontier experiments should involve the nano-space dynamics of nano-tori and Klein bottle graphenic structures, and their interconnected and entangled multi-spaces; that is, Equation (6) of the main text. This corresponds with mixing topological mathematics with the theories of quantum mechanical interpretation and observations, and particularly the theory of multi-verses [30,31,32]. Furthermore, the current context is a mathematical and computational movement aimed at unifying the big two physical theories—namely, quantum mechanics and general relativity—in terms of space mathematics and computational confinement and dynamics, as we referred to in the recently launched project of Wolfram Research Company; see [33].

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**Figure 1.**So-called “graphenic nano-structures”: (

**A**) the graphene graph as a 2D structure “with boundary”; (

**B**) the torus variety; and (

**C**) the Klein bottle variety, as the 2D manifolds without a boundary; see the text for more details regarding the Möbius strip, see [2].

**Figure 2.**Polyhex lattice with L

_{M}= 3, L

_{C}= 1. The unit cell is the usual C-shaped cell consisting of the four atoms: 1, 2, 3, and 4. This graphenic lattice G

_{3,1}has open bonds along y that are closed in a parallel manner along x forming a cylinder: node 2 bounds 9, and 3 bounds 12. The graphenic torus T

_{3,1}has the open bonds along x closed in a parallel manner along y: 1–4, 5–8, 9–12. By sewing the open bonds along x in the antiparallel manner (1–12, 5–8, 4–9), the Klein bottle KB

_{3,1}is built. Nodes 1, 4, 6, 7, 9, 10, 11, and 12 are topologically equivalent to {b

_{k}} = {3,5,3}, with shrunk eccentricity $\underset{\xaf}{\epsilon}$ = 3; the remaining vertices 2, 3, 5, and 8 share the coordination numbers {b

_{k}} = {3,4,3,1} and eccentricity $\overline{\epsilon}=4$ of the T

_{3,1}nodes.

**Figure 3.**Polyhex lattices with L

_{M}6, L

_{C}= 2. (

**A**) The toroidal fullerene T

_{(6,2)}is made of N = 48 equivalent nodes with $\overline{w}=98$. Edges are closed to form the torus by sewing the balls with the same color in a parallel manner; (

**B**) the Klein bottle KB

_{(6,2)}built by gluing, in an antiparallel manner, the nodes of the zigzag edge, such as 25–24, 29–20, ..., 45–4. Black circles have the same transmission of the torus’ vertices, whereas the remaining circles have lower shrunken values, making KB

_{(6,2)}topologically more compact than T

_{(6,2)}.

**Figure 4.**KB

_{3,2}graph with L

_{M}= 3, L

_{C}= 2. Open bonds along x are sewn in the antiparallel manner (13–12, 17–8, 21–4); open bonds along y are close, in parallel. This Klein bottle is formed by N = 24 equivalent nodes with {${b}_{k}$} = {3,6,8,5,1}, the same set shown by all nodes of the T

_{3,2}torus.

**Figure 5.**For L

_{M}= 5, Ł

_{C}= 4, and L

_{C}= 1, 2, …, 5, the topological roundness ${\rho}_{E}$ is plotted for both ${\mathrm{KB}}_{{L}_{M},{L}_{C}}$ and ${\mathrm{T}}_{{L}_{M},{L}_{C}}$ isomeric graphs; when L

_{C}≥ Ł

_{C}, the Klein bottle graphs are topologically equivalent to the tori ${\mathrm{T}}_{{L}_{M},{L}_{C}}$.

**Table 1.**Topological classes for the vertices of the graph $K{B}_{6,2}$, including eccentricity ${\u03f5}_{i}$, transmission ${w}_{i}$, {${b}_{ik}$} sets, Wiener number W, and extreme topological efficiency ${\rho}_{E}$; multiplicity is given in brackets. ${T}_{6,2}$ descriptors hold for all nodes of the toroidal polyhex and coincide with the last class of $K{B}_{6,2}$.

$K{B}_{6,2}$ W = 4504; ${\rho}_{E}$ = 98/91 = 1.0769 | |||

V | {${b}_{ik}$} | ${\u03f5}_{i}$ | ${w}_{i}$ |

(8) v5 v8 v17 v20 v29 v32 v41 v44 | 3 6 9 12 11 5 1 | 7 | 91 |

(16) v6 v7 v10 v11 v18 v19 v22 v23 v30 v31 v34 v35 v42 v43 v46 v47 | 3 6 9 11 11 6 1 | 7 | 92 |

(16) v1 v4 v9 v12 v13 v16 v21 v24 v25 v28 v33 v36 v37 v40 v45 v48 | 3 6 9 10 9 7 3 | 7 | 95 |

(8) v2 v3 v14 v15 v26 v27 v38 v39 | 3 6 9 9 8 7 4 1 | 8 | 98 |

${\mathrm{T}}_{6,2}$ W = 4704; ${\rho}_{E}$ = 1 | |||

(48) v1, v2, …, v47, v48 | 3 6 9 9 8 7 4 1 | 8 | 98 |

**Table 2.**For the family of graphs $K{B}_{3l,l}$, the eccentricity {${\u03f5}_{i}$} values are compared to the tori ${T}_{3l,l}$ eccentricity $\overline{\epsilon}$ for $l$ = 1,2,..,10. Indicating with $\underset{\_}{\mathit{\epsilon}}$

**=**min{${\u03f5}_{i}\}$, the topological shrinkage $\underset{\_}{\epsilon}<\overline{\epsilon}$ of the Klein bottles is a peculiar outcome of the current study.

$\mathit{l}$ | $\mathit{K}{\mathit{B}}_{3\mathit{l},\mathit{l}}\left\{{\mathit{\u03f5}}_{\mathit{i}}\right\}$$\underset{\xaf}{\mathit{\epsilon}}$ = $\mathbf{min}\left\{{\mathit{\u03f5}}_{\mathit{i}}\right\}$ | ${\mathit{T}}_{3\mathit{l},\mathit{l}}\overline{\mathit{\epsilon}}$ |
---|---|---|

1 | 3,4$\underset{\_}{\mathit{\epsilon}}$ = 3 | 4 |

2 | 7,8$\underset{\_}{\mathit{\epsilon}}$= 7 | 8 |

3 | 10,11,12$\underset{\_}{\mathit{\epsilon}}$= 10 | 12 |

4 | 13,14,15,16$\underset{\_}{\mathit{\epsilon}}$= 13 | 16 |

5 | 17,18,19,20$\underset{\_}{\mathit{\epsilon}}$= 17 | 20 |

6 | 20,21,22,23,24$\underset{\_}{\mathit{\epsilon}}$= 20 | 24 |

7 | 23,24,25,26,27,28$\underset{\_}{\mathit{\epsilon}}$= 23 | 28 |

8 | 27,28,29,30,31,32$\underset{\_}{\mathit{\epsilon}}$= 27 | 32 |

9 | 30,31,32,33,34,35,36$\underset{\_}{\mathit{\epsilon}}$= 30 | 36 |

10 | 33,34,35,36,37,38,39,40$\underset{\_}{\mathit{\epsilon}}$= 33 | 40 |

**Table 3.**For L

_{M}= 5,6, the eccentricities

**{**${\u03f5}_{i}\}$ of the ${\mathrm{KB}}_{{L}_{M},{L}_{C}}$ graphs are compared with the toroidal case, L

_{C}= 1, 2, …. Graphs with L

_{C}< L

_{M}− 1 show the topological shrinkage of the eccentricity $\underset{\_}{\epsilon}<\overline{\epsilon}$ and Wiener index W

_{KB}< W

_{T}; for L

_{C}≥ L

_{M}− 1, the KB graphs are topologically equivalent, $\underset{\_}{\epsilon}=\overline{\epsilon}$, to the tori ${\mathrm{T}}_{{L}_{M},{L}_{C}}$ of the same sizes with W

_{KB}= W

_{T}.

L_{M} = 5, Ł_{C} = 4; for L_{C} ≥ Ł_{C}, the two polyhexes are equivalent. | ||||

L_{C} | $K{B}_{\mathbf{5},{\mathit{L}}_{\mathit{C}}}${${\mathit{\u03f5}}_{\mathit{i}}$}$\underset{\xaf}{\mathit{\epsilon}}$= min{${\u03f5}_{i}$} | $\mathit{W}$_{KB} | ${T}_{\mathbf{5},{\mathit{L}}_{\mathit{C}}}\overline{\mathit{\epsilon}}$ | $\mathit{W}$_{T} |

1 | 4,5,6$\underset{\_}{\epsilon}$= 4 | 528 | 6 | 600 |

2 | 6,7$\underset{\_}{\epsilon}$= 6 | 2800 | 7 | 2880 |

3 | 7,8$\underset{\_}{\epsilon}$= 7 | 7632 | 8 | 7680 |

4 | 9$\underset{\_}{\epsilon}$= 9 | 16,000 | 9 | 16,000 |

5 | 10$\underset{\_}{\epsilon}$= 10 | 29,000 | 10 | 29,000 |

6 | 12$\underset{\_}{\epsilon}$= 12 | 48,000 | 12 | 48,000 |

7 | 14$\underset{\_}{\epsilon}$= 14 | 74,200 | 14 | 74,200 |

8 | 16$\underset{\_}{\epsilon}$= 16 | 108,800 | 16 | 108,800 |

9 | 18$\underset{\_}{\epsilon}$= 18 | 153,000 | 18 | 153,000 |

10 | 20$\underset{\_}{\epsilon}$= 20 | 208,000 | 20 | 208,000 |

15 | 30$\underset{\_}{\epsilon}$= 30 | 687,000 | 30 | 687,000 |

20 | 40$\underset{\_}{\epsilon}$= 40 | 1,616,000 | 40 | 1,616,000 |

25 | 50$\underset{\_}{\epsilon}$= 50 | 3,145,000 | 50 | 3,145,000 |

L_{M} = 6, Ł_{C} = 5; for L_{C} ≥ Ł_{C}, the two polyhexes are equivalent. | ||||

L_{C} | $K{B}_{\mathbf{6},{\mathit{L}}_{\mathit{C}}}${${\mathit{\u03f5}}_{\mathit{i}}$}$\underset{\xaf}{\mathit{\epsilon}}$= min{${\mathit{\u03f5}}_{\mathit{i}}$} | $\mathit{W}$_{KB} | ${T}_{\mathbf{6},{\mathit{L}}_{\mathit{C}}}\overline{\mathit{\epsilon}}$ | $\mathit{W}$_{T} |

1 | 4,5,6,7$\underset{\_}{\epsilon}$= 4 | 860 | 7 | 1008 |

2 | 7,8$\underset{\_}{\epsilon}$= 7 | 4504 | 8 | 4704 |

3 | 8,9$\underset{\_}{\epsilon}$= 8 | 12,084 | 9 | 12,240 |

4 | 9,10$\underset{\_}{\epsilon}$= 9 | 24,880 | 10 | 24,960 |

5 | 11$\underset{\_}{\epsilon}$= 11 | 44,400 | 11 | 44,400 |

6 | 12$\underset{\_}{\epsilon}$= 12 | 72,288 | 12 | 72,288 |

7 | 14$\underset{\_}{\epsilon}$= 14 | 110,544 | 14 | 110,544 |

8 | 16$\underset{\_}{\epsilon}$= 16 | 160,896 | 16 | 160,896 |

9 | 18$\underset{\_}{\epsilon}$= 18 | 225,072 | 18 | 225,072 |

10 | 20$\underset{\_}{\epsilon}$= 20 | 304,800 | 20 | 304,800 |

15 | 30$\underset{\_}{\epsilon}$= 30 | 997,200 | 30 | 997,200 |

20 | 40$\underset{\_}{\epsilon}$= 40 | 2,337,600 | 40 | 2,337,600 |

25 | 50$\underset{\_}{\epsilon}$= 50 | 4,542,000 | 50 | 4,542,000 |

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**MDPI and ACS Style**

Putz, M.V.; Ori, O.
Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems. *Symmetry* **2020**, *12*, 1233.
https://doi.org/10.3390/sym12081233

**AMA Style**

Putz MV, Ori O.
Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems. *Symmetry*. 2020; 12(8):1233.
https://doi.org/10.3390/sym12081233

**Chicago/Turabian Style**

Putz, Mihai V., and Ottorino Ori.
2020. "Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems" *Symmetry* 12, no. 8: 1233.
https://doi.org/10.3390/sym12081233