# A Symmetry Motivated Link Table

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Conjecture**

**1.**

#### Importance of Link Symmetries in DNA Topology

## 2. Writhe and Linking Number

## 3. Link Symmetries and Nomenclature

#### 3.1. Isotopy Classes

#### 3.2. Doll and Hoste Notation

#### 3.3. Link Symmetries

- L is purely invertible if it is isotopic to the link found by simultaneously reversing both components (L++ = L−−).
- L is fully invertible if it is isotopic to L with every other choice of orientation.
- L has even operations symmetry if it is isotopic to links obtained by an even number of reflections and/or component reversals.
- L has pure exchange symmetry if it is isotopic to L with the component labels exchanged (L++ = $\tau $L++).
- L has a non-pure exchange symmetry if it is isotopic to L with a combination of exchanged labels with a reflection and/or with component reversals, but L++ ≠ $\tau $L++.
- L has no exchange symmetry, if it is not isotopic to L with the component labels exchanged regardless of any reversals or reflections.
- L has full symmetry if it is isotopic to every link obtained by component relabeling, component reversal, and reflection.
- L has no symmetry if it is not isotopic to any link obtained by component relabeling, component reversal, or reflection.

#### 3.4. Symmetries, Writhe and Linking Number

#### 3.5. Previous Classification Schemes

## 4. Defining a Canonical Isotopy Class for Links

#### 4.1. Cubic Lattice Links and the BFACF Algorithm

#### 4.2. Canonical Isotopy Class

**Case 1, L is a knot ($c=1$)**

**Case 2, L is a 2-component link ($c=2$)**

#### 4.3. Proposed Link Table

#### 4.4. Note on Minimum Lattice Links

## 5. Results and Discussion

#### 5.1. Numerical Results

#### 5.2. Boundedness of Writhe under BFACF Moves

**Theorem**

**1.**

**Corollary**

**1.**

**Theorem**

**2.**

**Proof.**

- the BFACF edge runs from $(0,0,0)$ to $(0,1,0$), and
- the result of the BFACF move will push the BFACF edge to an edge from $(0,0,-1)$ to $(0,1,-1)$.

## 6. Numerical Methods

#### 6.1. BFACF Simulations

#### 6.2. Minimum Length Links

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Link Table

**Figure A1.**Regular oriented diagrams representing the link isotopy classes for L++ chosen as described in Section 4. Next to each link name is its symmetry group, which may be cross-referenced with Table 1. For links lacking pure exchange symmetry, the lighter blue strand is labeled as component 1 and the darker red-orange strand is component 2.

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**Figure 1.**(

**a**) Example of a coherent band surgery use to model DNA recombination; (

**b**) Contribution of each type of crossing to the projected writhe calculation.

**Figure 2.**Isotopy types of the link ${4}_{1}^{2}$ are pictured in the first column. Each row denotes a different isotopy type of the ${4}_{1}^{2}$ used as the starting point for a coherent band surgery (local reconnection). These can be interpreted as substrates of site-specific recombination at two dif sites, one on each component of the link. The right side shows all the potential products of said event, up to crossing number 6, depending on the isotopy type of the substrate. It is clear in this example that coherent band surgery on the different isotopy types of the link ${4}_{1}^{2}$ yields different knots. In particular, ${4}_{1}^{2}$+− and ${4}_{1}^{2*}$++ can be unknotted in one step, while ${4}_{1}^{2}$++ and ${4}_{1}^{2*}$+− cannot.

**Figure 3.**Example of the link notation adopted and modified from the work of Doll and Hoste [13]. The lighter blue strand is component 1 and the darker red-orange strand is component 2. Here, $\tau $ is the nontrivial element of ${S}_{2}$. Because ${4}_{1}^{2}$ has symmetry group ${\Sigma}_{4,1}$, all links sharing a row in this figure are equivalent. The diagram labeled ${4}_{1}^{2}$++ here matches the diagram in Figure A1. All other diagrams are determined from ${4}_{1}^{2}$++.

**Figure 4.**(

**a**) BFACF moves: ($\pm 2$)-move, top; ($+0$)-move, bottom; (

**b**) A minimum step cubic lattice representation of the ${8}_{15}^{2}$ link.

**Figure 5.**Estimates with 95% confidence intervals for ${\mathcal{S}}_{n}($L++) for a selection of links for $n\in \{76,100,150,250,\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}300\}$. Data were obtained from the simulations as described in Section 6. The expected value of ${\mathcal{S}}_{n}({8}_{15}^{2})$ was the lowest among links lacking reflection symmetry. Note that even though the expected value is relatively small for all lengths examined, the confidence intervals do not include 0. Limited variability of ${\mathcal{S}}_{n}($++) was observed as length increased for all prime links with up to 9 crossings. This suggests a well-behaved nature of writhe for long lattice links. Confidence intervals for all sampled links can be found in Supplementary Table S2. Values shown are for some of the isotopy classes in Figure A1.

**Figure 6.**If a BFACF move is performed on the black edge in the direction of the orange (medium gray in grayscale) edge of the push-off beneath it, then the linking number with the push-off will change by $-1$ which will change the writhe by $-1/4$. This same BFACF move will also push the blue (dark gray) edge of the push-off through the yellow (light gray) edge of the link, which will cause the linking number to change by another $-1$, hence this will contribute a $-1/4$ change to the writhe. Thus, a BFACF move pushing the black edge into the page will result in a lattice link with a writhe $1/2$ less than the current link’s writhe.

**Figure 7.**(

**a**) This graph shows 95% confidence intervals for ${\mathcal{S}}_{n}($L++) of the four links with reflection symmetry and crossing number up to 9, for lengths 76, 100, 150, 200, 250, and 300; (

**b**) This graph illustrates the expected behavior of ${\mathcal{S}}_{n}(L,i)$ for a link with pure exchange symmetry (${7}_{2}^{2}$) and a link without pure exchange symmetry (${8}_{15}^{2}$), where i denotes the component number. The large error bars are due to a smaller sample size for ${7}_{2}^{2}$ at length 200; however, even for low sample sizes, the error bars overlap as expected.

**Table 1.**Symmetry groups for two-component links with up to nine crossings. Listed are names for the groups and their notation as a subgroup of ${\Gamma}_{2}$ [14,15]. Also listed are generators for the subgroup where $\u03f5$ is a reflection, ${r}_{1}$ and ${r}_{2}$ are reversals of components 1 and 2, respectively, and p is the exchange of the component labels. The final column indicates which of the 16 different possible isotopy classes are equivalent to L++. Here $\tau $ is the non-trivial element of ${S}_{2}$.

Symmetry Name | Occurences for $\mathit{c}(\mathit{L})\le 9$ | Subgroup of ${\mathbf{\Gamma}}_{2}$ | Generators of Subgroup | Equivalence Class of $\mathit{L}++$ |
---|---|---|---|---|

Full Symmetry | 1 | ${\Gamma}_{2}$ | $\langle \u03f5,{r}_{1},{r}_{2},p\rangle $ | $\{L$++, L+−, L−+, L−−, ${L}^{*}$++,${L}^{*}$+−, ${L}^{*}$−+, ${L}^{*}$−−, $\tau $L++, $\tau $L+−,$\tau $L−+, $\tau $L−−, $\tau $${L}^{*}$++, $\tau $${L}^{*}$+−,$\tau $${L}^{*}$−+, $\tau $${L}^{*}$−−} |

Purely Inv. (Pure Ex.) | 25 | ${\Sigma}_{4,1}$ | $\langle {r}_{1}{r}_{2},p\rangle $ | $\{L$++$,L$−−, $\tau $L++, $\tau $L−−} |

Purely Inv. (No Ex.) | 32 | ${\Sigma}_{2,1}$ | $\langle {r}_{1}{r}_{2}\rangle $ | $\{L$++$,L$−−} |

Fully Inv. (Pure Ex.) | 5 | ${\Sigma}_{8,1}$ | $\langle {r}_{1},{r}_{2},p\rangle $ | $\{L$++$,L$+−$,L$−+$,L$−−,$\tau $L++, $\tau $L+−, $\tau $L−+, $\tau $L−−} |

Fully Inv. (no Ex.) | 22 | ${\Sigma}_{4,2}$ | $\langle {r}_{1},{r}_{2}\rangle $ | $\{L$++$,L$+−$,L$−+$,L$−−} |

Even Op. (Pure Ex.) | 3 | ${\Sigma}_{8,2}$ | $\langle \u03f5{r}_{1},\u03f5{r}_{2},p\rangle $ | $\{L$++, L−−, ${L}^{*}$+−, ${L}^{*}$−+, $\tau $L++,$\tau $L−−, $\tau $${L}^{*}$+−, $\tau $${L}^{*}$−+} |

Even Op. (Non-Pure Ex.) | 1 | ${\Sigma}_{4,5}$ | $\langle \u03f5{r}_{1}p,\u03f5{r}_{2}p\rangle $ | $\{L$++$,L$−−$,\tau {L}^{*}$+−$,\tau {L}^{*}$−+} |

No Symmetry | 3 | $\left\{e\right\}$ | $\langle e\rangle $ | $\{L$++} |

**Table 2.**Columns 2, 3, and 4 show confidence intervals for the average of the sum of self-writhes (${\mathcal{S}}_{200}(L$++)), and self-writhes of components 1 and 2 (${\mathcal{S}}_{200}(L$++$,1)$ and ${\mathcal{S}}_{200}(L$++$,2)$) for length 200 links in ${\mathbb{Z}}^{3}$. For each 2-component link indicated in column 1, the average is taken over an ensemble of statistically independent length 200 lattice links of type L as described in the numerical methods section. Combined with the linking number (column 7), these confidence intervals are used to determine which diagram appears as L++ in Figure A1. The Rolfsen ([2]) diagram’s designation under our notation is presented in column 5. Column 6 lists which isotopy class is represented by default KnotPlot. Note that the KnotPlot conformations are reflections of those in the Rolfsen table. Symmetry groups (column 8) are taken from the work of Henry and Weeks, Berglund et al., and from SnapPy [14,16,22].

L | ${\mathcal{S}}_{200}(\mathit{L})$ | ${\mathcal{S}}_{200}(\mathit{L},1)$ | ${\mathcal{S}}_{200}(\mathit{L},2)$ | Rolfsen | KP | lk$(\mathit{L})$ | Sym |
---|---|---|---|---|---|---|---|

${0}_{1}^{2}$ | [$-\phantom{\rule{1.em}{0ex}}-$] | [$-\phantom{\rule{1.em}{0ex}}-$] | [$-\phantom{\rule{1.em}{0ex}}-$] | ${0}_{1}^{2}$ | ${0}_{1}^{2}$++ | 0 | ${\Gamma}_{2}$ |

${2}_{1}^{2}$ | [$-0.037\phantom{\rule{1.em}{0ex}}0.102$] | [$-0.054\phantom{\rule{1.em}{0ex}}0.043$] | [$-0.106\phantom{\rule{1.em}{0ex}}0.087$] | ${2}_{1}^{2}$ | ${2}_{1}^{2}$++ | 1 | ${\Sigma}_{8,2}$ |

${4}_{1}^{2}$ | [$0.755\phantom{\rule{1.em}{0ex}}0.877$] | [$0.391\phantom{\rule{1.em}{0ex}}0.481$] | [$0.336\phantom{\rule{1.em}{0ex}}0.424$] | ${4}_{1}^{2}$${}^{*}$ | ${4}_{1}^{2}$+− | 2 | ${\Sigma}_{4,1}$ |

${5}_{1}^{2}$ | [$1.401\phantom{\rule{1.em}{0ex}}1.607$] | [$0.685\phantom{\rule{1.em}{0ex}}0.844$] | [$0.657\phantom{\rule{1.em}{0ex}}0.822$] | ${5}_{1}^{2}$${}^{*}$ | ${5}_{1}^{2}$++ | 0 | ${\Sigma}_{8,1}$ |

${6}_{1}^{2}$ | [$1.624\phantom{\rule{1.em}{0ex}}1.692$] | [$0.812\phantom{\rule{1.em}{0ex}}0.862$] | [$0.795\phantom{\rule{1.em}{0ex}}0.847$] | ${6}_{1}^{2}$${}^{*}$ | ${6}_{1}^{2}$++ | 3 | ${\Sigma}_{4,1}$ |

${6}_{2}^{2}$ | [$-0.156\phantom{\rule{1.em}{0ex}}0.042$] | [$-0.144\phantom{\rule{1.em}{0ex}}0.014$] | [$-0.067\phantom{\rule{1.em}{0ex}}0.083$] | ${6}_{2}^{2}$ | ${6}_{2}^{2}$++ | 3 | ${\Sigma}_{8,2}$ |

${6}_{3}^{2}$ | [$1.957\phantom{\rule{1.em}{0ex}}2.225$] | [$0.979\phantom{\rule{1.em}{0ex}}1.207$] | [$0.892\phantom{\rule{1.em}{0ex}}1.105$] | ${6}_{3}^{2}$${}^{*}$ | ${6}_{3}^{2}$+− | 2 | ${\Sigma}_{4,1}$ |

${7}_{1}^{2}$ | [$2.188\phantom{\rule{1.em}{0ex}}2.364$] | [$1.027\phantom{\rule{1.em}{0ex}}1.167$] | [$1.109\phantom{\rule{1.em}{0ex}}1.249$] | ${7}_{1}^{2}$${}^{*}$ | ${7}_{1}^{2}$++ | 1 | ${\Sigma}_{4,1}$ |

${7}_{2}^{2}$ | [$0.413\phantom{\rule{1.em}{0ex}}0.788$] | [$0.211\phantom{\rule{1.em}{0ex}}0.509$] | [$0.093\phantom{\rule{1.em}{0ex}}0.389$] | ${7}_{2}^{2}$${}^{*}$ | ${7}_{2}^{2}$+− | 1 | ${\Sigma}_{4,1}$ |

${7}_{3}^{2}$ | [$2.667\phantom{\rule{1.em}{0ex}}2.728$] | [$1.318\phantom{\rule{1.em}{0ex}}1.373$] | [$1.324\phantom{\rule{1.em}{0ex}}1.38$] | ${7}_{3}^{2}$${}^{*}$ | ${7}_{3}^{2}$++ | 0 | ${\Sigma}_{8,1}$ |

${7}_{4}^{2}$ | [$4.292\phantom{\rule{1.em}{0ex}}4.348$] | [$3.992\phantom{\rule{1.em}{0ex}}4.04$] | [$0.289\phantom{\rule{1.em}{0ex}}0.319$] | ${7}_{4}^{2}$ | ${7}_{4}^{2}$${}^{*}+$+ | 0 | ${\Sigma}_{4,2}$ |

${7}_{5}^{2}$ | [$2.532\phantom{\rule{1.em}{0ex}}2.602$] | [$2.843\phantom{\rule{1.em}{0ex}}2.904$] | [$-0.326\phantom{\rule{1.em}{0ex}}-0.286$] | ${7}_{5}^{2}$${}^{*}$ | $\tau $${7}_{5}^{2}$++ | 2 | ${\Sigma}_{2,1}$ |

${7}_{6}^{2}$ | [$1.411\phantom{\rule{1.em}{0ex}}1.445$] | [$1.381\phantom{\rule{1.em}{0ex}}1.41$] | [$0.023\phantom{\rule{1.em}{0ex}}0.042$] | ${7}_{6}^{2}$${}^{*}$ | $\tau $${7}_{6}^{2}$++ | 0 | ${\Sigma}_{4,2}$ |

${7}_{7}^{2}$ | [$3.51\phantom{\rule{1.em}{0ex}}3.592$] | [$3.458\phantom{\rule{1.em}{0ex}}3.527$] | [$0.036\phantom{\rule{1.em}{0ex}}0.081$] | ${7}_{7}^{2}$ | ${7}_{7}^{2}$${}^{*}+$+ | 2 | ${\Sigma}_{2,1}$ |

${7}_{8}^{2}$ | [$3.248\phantom{\rule{1.em}{0ex}}3.324$] | [$3.304\phantom{\rule{1.em}{0ex}}3.368$] | [$-0.07\phantom{\rule{1.em}{0ex}}-0.03$] | ${7}_{8}^{2}$${}^{*}$ | $\tau $${7}_{8}^{2}$++ | 0 | ${\Sigma}_{4,2}$ |

${8}_{1}^{2}$ | [$2.443\phantom{\rule{1.em}{0ex}}2.477$] | [$1.225\phantom{\rule{1.em}{0ex}}1.251$] | [$1.209\phantom{\rule{1.em}{0ex}}1.235$] | ${8}_{1}^{2}$${}^{*}$ | ${8}_{1}^{2}$+− | 4 | ${\Sigma}_{4,1}$ |

${8}_{2}^{2}$ | [$0.761\phantom{\rule{1.em}{0ex}}0.79$] | [$0.373\phantom{\rule{1.em}{0ex}}0.396$] | [$0.379\phantom{\rule{1.em}{0ex}}0.403$] | ${8}_{2}^{2}$ | ${8}_{2}^{2}$${}^{*}+$+ | 4 | ${\Sigma}_{4,1}$ |

${8}_{3}^{2}$ | [$2.861\phantom{\rule{1.em}{0ex}}2.907$] | [$1.435\phantom{\rule{1.em}{0ex}}1.474$] | [$1.41\phantom{\rule{1.em}{0ex}}1.449$] | ${8}_{3}^{2}$${}^{*}$ | ${8}_{3}^{2}$++ | 3 | ${\Sigma}_{4,1}$ |

${8}_{4}^{2}$ | [$0.868\phantom{\rule{1.em}{0ex}}0.94$] | [$0.417\phantom{\rule{1.em}{0ex}}0.476$] | [$0.429\phantom{\rule{1.em}{0ex}}0.486$] | ${8}_{4}^{2}$${}^{*}$ | ${8}_{4}^{2}$+− | 4 | ${\Sigma}_{4,1}$ |

${8}_{5}^{2}$ | [$1.171\phantom{\rule{1.em}{0ex}}1.22$] | [$0.575\phantom{\rule{1.em}{0ex}}0.618$] | [$0.578\phantom{\rule{1.em}{0ex}}0.62$] | ${8}_{5}^{2}$${}^{*}$ | ${8}_{5}^{2}$++ | 3 | ${\Sigma}_{4,1}$ |

${8}_{6}^{2}$ | [$3.29\phantom{\rule{1.em}{0ex}}3.33$] | [$1.632\phantom{\rule{1.em}{0ex}}1.671$] | [$1.639\phantom{\rule{1.em}{0ex}}1.678$] | ${8}_{6}^{2}$${}^{*}$ | ${8}_{6}^{2}$++ | 2 | ${\Sigma}_{4,1}$ |

${8}_{7}^{2}$ | [$2.829\phantom{\rule{1.em}{0ex}}2.864$] | [$1.415\phantom{\rule{1.em}{0ex}}1.445$] | [$1.402\phantom{\rule{1.em}{0ex}}1.431$] | ${8}_{7}^{2}$${}^{*}$ | ${8}_{7}^{2}$+− | 1 | ${\Sigma}_{4,1}$ |

${8}_{8}^{2}$ | [$-0.002\phantom{\rule{1.em}{0ex}}0.033$] | [$0.006\phantom{\rule{1.em}{0ex}}0.035$] | [$-0.02\phantom{\rule{1.em}{0ex}}0.01$] | ${8}_{8}^{2}$ | ${8}_{8}^{2}$++ | 1 | ${\Sigma}_{8,2}$ |

${8}_{9}^{2}$ | [$0.777\phantom{\rule{1.em}{0ex}}0.92$] | [$0.471\phantom{\rule{1.em}{0ex}}0.599$] | [$0.276\phantom{\rule{1.em}{0ex}}0.35$] | ${8}_{9}^{2}$ | $\tau $${8}_{9}^{2}$${}^{*}+$+ | 2 | ${\Sigma}_{2,1}$ |

${8}_{10}^{2}$ | [$0.893\phantom{\rule{1.em}{0ex}}0.927$] | [$0.594\phantom{\rule{1.em}{0ex}}0.625$] | [$0.291\phantom{\rule{1.em}{0ex}}0.309$] | ${8}_{10}^{2}$ | ${8}_{10}^{2}$${}^{*}+$+ | 0 | ${\Sigma}_{4,2}$ |

${8}_{11}^{2}$ | [$4.944\phantom{\rule{1.em}{0ex}}4.98$] | [$4.423\phantom{\rule{1.em}{0ex}}4.455$] | [$0.512\phantom{\rule{1.em}{0ex}}0.534$] | ${8}_{11}^{2}$ | ${8}_{11}^{2}$${}^{*}+$+ | 2 | ${\Sigma}_{2,1}$ |

${8}_{12}^{2}$ | [$1.904\phantom{\rule{1.em}{0ex}}1.932$] | [$2.422\phantom{\rule{1.em}{0ex}}2.447$] | [$-0.526\phantom{\rule{1.em}{0ex}}-0.508$] | ${8}_{12}^{2}$${}^{*}$ | ${8}_{12}^{2}$++ | 0 | ${\Sigma}_{4,2}$ |

${8}_{13}^{2}$ | [$1.945\phantom{\rule{1.em}{0ex}}2.0$] | [$1.917\phantom{\rule{1.em}{0ex}}1.965$] | [$0.018\phantom{\rule{1.em}{0ex}}0.046$] | ${8}_{13}^{2}$${}^{*}$ | ${8}_{13}^{2}$++ | 0 | ${\Sigma}_{4,2}$ |

${8}_{14}^{2}$ | [$3.13\phantom{\rule{1.em}{0ex}}3.185$] | [$3.17\phantom{\rule{1.em}{0ex}}3.218$] | [$-0.051\phantom{\rule{1.em}{0ex}}-0.022$] | ${8}_{14}^{2}$ | ${8}_{14}^{2}$${}^{*}+$+ | 2 | ${\Sigma}_{2,1}$ |

${8}_{15}^{2}$ | [$0.029\phantom{\rule{1.em}{0ex}}0.056$] | [$0.034\phantom{\rule{1.em}{0ex}}0.048$] | [$-0.01\phantom{\rule{1.em}{0ex}}0.014$] | ${8}_{15}^{2}$ | $\tau $${8}_{15}^{2}$${}^{*}+$+ | 0 | ${\Sigma}_{4,2}$ |

${8}_{16}^{2}$ | [$0.188\phantom{\rule{1.em}{0ex}}0.219$] | [$0.132\phantom{\rule{1.em}{0ex}}0.16$] | [$0.05\phantom{\rule{1.em}{0ex}}0.065$] | ${8}_{16}^{2}$ | ${8}_{16}^{2}$${}^{*}+$+ | 2 | ${\Sigma}_{2,1}$ |

**Table 3.**Mean self-writhes of minimum step prime 2-component links with 8 or fewer crossings. Numbers based on all conformations found in the preprint by Freund et al. [25].

Link | ${\mathcal{S}}_{min}(\mathit{L})$ | ${\mathcal{S}}_{min}(\mathit{L},1)$ | ${\mathcal{S}}_{min}(\mathit{L},2)$ |
---|---|---|---|

${0}_{1}^{2}$ | $0.0$ | $0.0$ | $0.0$ |

${2}_{1}^{2}$ | $0.0$ | $0.0$ | $0.0$ |

${4}_{1}^{2}$ | $0.8125$ | $0.4063$ | $0.4063$ |

${5}_{1}^{2}$ | $1.3492$ | $0.6746$ | $0.6746$ |

${6}_{1}^{2}$ | $1.65$ | $0.825$ | $0.825$ |

${6}_{2}^{2}$ | $0.0$ | $0.0$ | $0.0$ |

${6}_{3}^{2}$ | $1.9438$ | $0.9719$ | $0.9719$ |

${7}_{1}^{2}$ | $2.1636$ | $1.0818$ | $1.0818$ |

${7}_{2}^{2}$ | $0.7$ | $0.35$ | $0.35$ |

${7}_{3}^{2}$ | $2.4903$ | $2.4427$ | $0.0476$ |

${7}_{4}^{2}$ | $4.2553$ | $4.1811$ | $0.0743$ |

${7}_{5}^{2}$ | $2.3625$ | $2.7563$ | $-0.3937$ |

${7}_{6}^{2}$ | $1.4375$ | $1.4375$ | $0.0$ |

${7}_{7}^{2}$ | $3.5368$ | $3.5368$ | $0.0$ |

${7}_{8}^{2}$ | $3.0479$ | $3.0479$ | $0.0$ |

${8}_{1}^{2}$ | $2.597$ | $1.2985$ | $1.2985$ |

${8}_{2}^{2}$ | $0.8123$ | $0.4062$ | $0.4062$ |

${8}_{3}^{2}$ | $2.7172$ | $1.3586$ | $1.3586$ |

${8}_{4}^{2}$ | $0.8164$ | $0.4082$ | $0.4082$ |

${8}_{5}^{2}$ | $1.1765$ | $0.5883$ | $0.5883$ |

${8}_{6}^{2}$ | $3.1666$ | $1.5833$ | $1.5833$ |

${8}_{7}^{2}$ | $2.7215$ | $1.3607$ | $1.3607$ |

${8}_{8}^{2}$ | $0.0$ | $0.0$ | $0.0$ |

${8}_{9}^{2}$ | $0.9355$ | $0.6021$ | $0.3333$ |

${8}_{10}^{2}$ | $0.9525$ | $0.7288$ | $0.2236$ |

${8}_{11}^{2}$ | $4.6944$ | $4.6667$ | $0.0278$ |

${8}_{12}^{2}$ | $2.0538$ | $2.2909$ | $-0.237$ |

${8}_{13}^{2}$ | $2.1324$ | $2.1324$ | $0.0$ |

${8}_{14}^{2}$ | $3.0967$ | $3.0967$ | $0.0$ |

${8}_{15}^{2}$ | $0.2157$ | $0.0$ | $0.2157$ |

${8}_{16}^{2}$ | $0.5$ | $0.5$ | $0.0$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Witte, S.; Flanner, M.; Vazquez, M. A Symmetry Motivated Link Table. *Symmetry* **2018**, *10*, 604.
https://doi.org/10.3390/sym10110604

**AMA Style**

Witte S, Flanner M, Vazquez M. A Symmetry Motivated Link Table. *Symmetry*. 2018; 10(11):604.
https://doi.org/10.3390/sym10110604

**Chicago/Turabian Style**

Witte, Shawn, Michelle Flanner, and Mariel Vazquez. 2018. "A Symmetry Motivated Link Table" *Symmetry* 10, no. 11: 604.
https://doi.org/10.3390/sym10110604