# The 27 Possible Intrinsic Symmetry Groups of Two-Component Links

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## Abstract

**:**

## 1. Introduction

`SnapPea`; see Section 3 for details. We present in Table 2 a report on how frequently each symmetry group occurs among the 77,036 two-component hyperbolic links with 14 or fewer crossings. In Section 5, we restrict to the case of alternating, nonsplit two-component links, for which only 12 of the 27 symmetry groups are possible. In particular, no alternating link with an even number of components may have full symmetry. We realize alternating, prime examples for 11 of these 12 groups.

## 2. The Whitten Group ${\Gamma}_{2}$

**Definition**

**2.1**

#### 2.1. Intrinsic Symmetries of a Link

- Permuting the ${K}_{i}$.
- Reversing the orientation of any set of ${K}_{i}$’s.
- Reversing the orientation on ${S}^{3}$ (mirroring L ).

**Definition**

**2.2**

**Example**

**2.3**

**Example**

**2.4**

#### 2.2. Notation

#### 2.3. Subgroups of ${\Gamma}_{2}$

**Proposition**

**2.5**

`Magma`. □

## 3. Computational Examples

`SnapPea`. This software can calculate the mapping class group $MCG({S}^{3}\backslash L)$ of the link complement. The elements of $MCG({S}^{3},L)$ which extend through the boundary tori to automorphisms of all of ${S}^{3}$ form a copy of $MCG({S}^{3},L)$ inside $MCG({S}^{3}\backslash L)$. We can detect such maps using the following standard lemma.

**Lemma**

**3.1**

`Python`front end

`SnapPy`for

`SnapPea`written by Marc Culler and Nathan Dunfield. To each map on the boundary tori of the link complement,

`SnapPy`assigns a collection of $\mu $ matrices along with a permutation element which records how the components of the link were permuted. Each matrix is $2\times 2$ and records the images of the meridians and longitudes of the appropriate component, along with the orientation of the ambient space. The effect of the map on the orientation ${\u03f5}_{i}$ of the given component and the orientation ${\u03f5}_{0}$ of ${S}^{3}$ is given by the rules below. Note that if the matrix for one boundary torus indicates that the orientation on ${S}^{3}$ is reversed, then so will the matrices for all other boundary tori, since these matrices result from restricting a single map on ${S}^{3}$.

`SnapPy`. We computed the symmetry group for all 77,036 two-component hyperbolic links of 14 and fewer crossings in

`SnapPy`’s database. Table 2 shows the census of symmetry groups of hyperbolic links found using

`SnapPy`. The data file containing the Whitten group elements for these links is included in the

`Arxiv`data repository and it is a future project to incorporate our computational techniques into

`SnapPy`.

`Mathematica`package

`KnotTheory`[8]. For each link L and each Whitten group element $\gamma $, we computed a number of invariants of L and ${L}^{\gamma}$ in an attempt to rule out $\gamma $ as an element of $\Sigma \left(L\right)$. To rule out “exchange” symmetries, we applied two tests. First, we computed the Jones polynomials of each component of L to try to rule out exchanges between components of different knot types. If that failed, we turned to the “satellite lemma”, which we use often in [6].

**Lemma**

**3.2**

`Mathematica`. Those symmetries which extended to symmetries of the link after crossing information was taken into account were classified according to the Whitten element they represented. We called these diagrammatic symmetries of each link. Of course, different diagrams are expected to reveal additional symmetries, so the diagrammatic symmetry group of a diagram of a link is only a subgroup of $\Sigma \left(L\right)$. In most cases, we still have a number of potential symmetries which may or may not be present for the link. However, we found 70 cases where the diagrammatic symmetry groups represented all of $\Sigma \left(L\right)$ since they agreed with the “supergroups” computed earlier: 6 links with $\Sigma \left(L\right)={\Sigma}_{4,1}$, 4 links with $\Sigma \left(L\right)={\Sigma}_{4,2}$, and 60 links with $\Sigma \left(L\right)={\Sigma}_{2,1}$.

## 4. Examples of Links with Particular Symmetry Groups

`SnapPea`and are new to the literature.

**Theorem**

**4.1**

## 5. Intrinsic Symmetry Groups of Alternating Links

**Lemma**

**5.1**

**Proposition**

**5.2**

## 6. Future Directions

**Conjecture**

**6.1**

`SnapPy`, searching for examples and developing a catalog of frequencies seems like an approachable problem.

## Acknowledgements

`Magma`computations of subgroups of ${\Gamma}_{\mu}$. Our work was supported by the UGA VIGRE grants DMS-07-38586 and DMS-00-89927 and by the UGA REU site grant DMS-06-49242.

## References

- Boileau, M.; Zimmermann, B. Symmetries of nonelliptic Montesinos links. Math. Ann.
**1987**, 277, 563–584. [Google Scholar] - Bonahon, F.; Siebenmann, L. New Geometric Splittings of Classical Knots, and the Classification and Symmetries of Arborescent Knots. http://www-bcf.usc.edu/fbonahon/Research/Preprints/Preprints.html (accessed on 17 February 2012).
- Henry, S.R.; Weeks, J.R. Symmetry groups of hyperbolic knots and links. J. Knot Theory Ramif.
**1992**, 1, 185–201. [Google Scholar] - Kodama, K.; Sakuma, M. Symmetry Groups of Prime Knots up to 10 Crossings. In Knots 90 (Osaka, 1990); de Gruyter: Berlin, Germany, 1992; pp. 323–340. [Google Scholar]
- Whitten, W.C., Jr. Symmetries of links. Trans. Am. Math. Soc.
**1969**, 135, 213–222. [Google Scholar] - Berglund, M.; Cantarella, J.; Casey, M.P.; Dannenberg, E.; George, W.; Johnson, A.; Kelly, A.; LaPointe, A.; Mastin, M.; Parsley, J.; Rooney, J.; Whitaker, R. Intrinsic symmetry groups of links with 8 and fewer crossings. Symmetry
**2012**, in press. [Google Scholar] - Hillman, J.A. Symmetries of knots and links, and invariants of abelian coverings. I. Kobe J. Math.
**1986**, 3, 7–27. [Google Scholar] - Bar-Natan, D.
`KnotTheory`package for`Mathematica`. http://katlas.org/wiki/The_Mathematica_Package_KnotTheory (accessed on 9 February 2012). - Conway, J.H. An Enumeration of Knots and Links, and Some of Their Algebraic Properties. In Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967); Pergamon: Oxford, UK, 1970; pp. 329–358. [Google Scholar]

**Figure 1.**The 27 subgroups (up to conjugacy) of the Whitten group ${\Gamma}_{2}$ are depicted below as a subgroup lattice. There are 8 pairs of conjugate subgroups of ${\Gamma}_{2}$; above, one representative is displayed for each conjugate pair; these are denoted by ‗. For 21 of these subgroups, we provide an example realizing this symmetry group in Table 1. The 6 subgroups for which no example has been found are depicted with a red dashed border.

**Figure 2.**The link ${7}_{6}^{2}$ (or $7a1$ in Thistlethwaite’s notation) has two unknotted components with linking number zero. To decide whether this link admits a pure exchange symmetry, we construct two satellite links. The center link has one component replaced by a Hopf link while the other is unframed, while the right link has the other component replaced by a Hopf link. If the original ${7}_{6}^{2}$ admits a pure exchange symmetry, these satellites would be isotopic. However the Jones polynomial of the center link is ${a}^{10}-2{a}^{9}+{a}^{8}-{a}^{6}+2{a}^{5}-{a}^{4}-\frac{1}{{a}^{4}}+2{a}^{3}+\frac{1}{{a}^{3}}+\frac{1}{{a}^{2}}-\frac{1}{a}+2$ while the Jones polynomial of the right-hand link is ${a}^{7}-2{a}^{6}+2{a}^{5}-2{a}^{4}+2{a}^{3}-\frac{1}{{a}^{3}}+{a}^{2}+\frac{2}{{a}^{2}}-\frac{2}{a}+3$. This proves that ${7}_{6}^{2}$ does not admit a pure exchange symmetry.

**Figure 3.**The 194 th non-hyperbolic two-component link with 14 crossings in Thistlethwaite’s table has $\Sigma \left(L\right)<{\Sigma}_{8,1}$ according to our polynomial and satellite lemma calculations. The original diagram of this link is shown at left. Since ${\Sigma}_{4,4}<{\Sigma}_{8,1}$, this link could have the “missing” symmetry group ${\Sigma}_{4,4}$. However, rearranging the diagram as shown on the right reveals that this link has pure exchange symmetry. Since pure exchange is not part of ${\Sigma}_{4,4}$, we see that this link cannot be an example of a link with ${\Sigma}_{4,4}$ symmetry.

**Figure 4.**Three new example links for symmetry groups, not found using

`SnapPea`. From left to right, these links exhibit the symmetry groups $\langle \rho \rangle $, $\langle m\rho \rangle $ and ${\Sigma}_{8,7}$.

**Figure 5.**An isotopy diagram showing that link 10n59 admits symmetry ${i}_{2}$—we may reverse the orientation of the second component.

**Table 1.**Examples of prime, nontrivial two-component links are currently known for 21 of the possible 27 intrinsic symmetry groups. Links are specified using Thistlethwaite notation and DT codes, as described in Section 2.2.

Symmetry Group | Example |
---|---|

$\left\{id\right\}$ | 11a164 |

${\Sigma}_{2,2}=\langle m\rangle $ | DT[8,10,−16:12,14,24,2,−20,−22,−4,−6,−18] |

${\Sigma}_{2,3}=\langle mPI\rangle $ | DT[8,10,−14:12,22,24,−18,−20,−4,−6,−16,2] |

${\Sigma}_{2,1}=\langle PI\rangle $ | 7a2 |

${\Sigma}_{2,4}=\langle {i}_{1}\rangle $ | 10a98 |

${\Sigma}_{2,5}=\langle \rho \rangle $ | DT[14,16,44,20,4,2,18,10,12,8,28:36,38,22,42,26,24,40,32,34,30,6] |

${\Sigma}_{2,6}=\langle m\rho \rangle $ | DT[44,−20,−16,−22,−14,−4,−6,−10,−12,−28,−8:2,34,−18,24,42,26,40,30,32,36,38] |

${\Sigma}_{2,7}=\langle m{i}_{1}\rangle $ | 10a81 |

${\Sigma}_{4,2}$ | 7a3 |

${\Sigma}_{4,4}$ | No example known |

${\Sigma}_{4,1}$ | 4a1 |

${\Sigma}_{4,11}$ | DT[16,−6,−12,−20,−22,−4,28:2,14,26,−10,−8,18,24] |

${\Sigma}_{4,3}$ | 10n46, 10a56 |

${\Sigma}_{4,5}$ | DT[14,6,10,16,4,20:22,8,2,24,12,18] |

${\Sigma}_{4,9}$ | 10n36 |

${\Sigma}_{4,6}$ | No example known |

${\Sigma}_{4,7}$ | No example known |

${\Sigma}_{4,8}$ | DT[10,−14,−18,24:2,28,−4,−12,−20,−6,−16,26,8,22] |

${\Sigma}_{4,10}$ | DT[10,−14,−20,24:2,28,−16,−4,−12,−6,−18,26,8,22] |

${\Sigma}_{8,7}$ | 10n59 |

${\Sigma}_{8,2}$ | 2a1 |

${\Sigma}_{8,1}$ | 5a1 |

${\Sigma}_{8,4}$ | DT[14,−16,22,−28,24,−18:12,−20,−10,−2,26,8,5,−6] |

${\Sigma}_{8,6}$ | No example known |

${\Sigma}_{8,3}$ | No example known |

${\Sigma}_{8,5}$ | DT[10,−14,−18,22:2,24,−4,−12,−6,−16,8,10] |

${\Gamma}_{2}$ | No example known |

**Table 2.**The number of links admitting each symmetry group among the 77,036 hyperbolic two-component links included in the Thistlethwaite table distributed with

`SnapPy`. Among these links, almost two-thirds had no symmetry (trivial symmetry group) and there were no examples with full symmetry.

Group | Number of Links | Group | Number of Links | Group | Number of Links |
---|---|---|---|---|---|

$\left\{id\right\}$ | 53,484 | ${\Sigma}_{4,1}$ | 1,396 | ${\Sigma}_{8,1}$ | 52 |

${\Sigma}_{4,2}$ | 2,167 | ${\Sigma}_{8,2}$ | 25 | ||

${\Sigma}_{2,1}$ | 17,951 | ${\Sigma}_{4,3}$ | 24 | ${\Sigma}_{8,3}$ | 0 |

${\Sigma}_{2,2}$ | 7 | ${\Sigma}_{4,4}$ | 0 | ${\Sigma}_{8,4}$ | 1 |

${\Sigma}_{2,3}$ | 9 | ${\Sigma}_{4,5}$ | 12 | ${\Sigma}_{8,5}$ | 2 |

${\Sigma}_{2,4}$ | 1,336 | ${\Sigma}_{4,6}$ | 0 | ${\Sigma}_{8,6}$ | 0 |

${\Sigma}_{2,5}$ | 418 | ${\Sigma}_{4,7}$ | 0 | ${\Sigma}_{8,7}$ | 8 |

${\Sigma}_{2,6}$ | 3 | ${\Sigma}_{4,8}$ | 2 | ||

${\Sigma}_{2,7}$ | 123 | ${\Sigma}_{4,9}$ | 11 | ${\Gamma}_{2}$ | 0 |

${\Sigma}_{4,10}$ | 1 | ||||

${\Sigma}_{4,11}$ | 4 |

**Table 3.**The number of subgroups of ${\Gamma}_{\mu}$, as computed by

`Magma`. Each one represents a different intrinsic symmetry group possible for a $\mu $-component link.

μ | $\left(\right)open="|"\; close="|">{\mathbf{\Gamma}}_{\mathit{\mu}}$ | # subgroups | # subgroups (up to conjugacy) |
---|---|---|---|

1 | 4 | 5 | 5 |

2 | 16 | 35 | 27 |

3 | 96 | 420 | 131 |

4 | 768 | 9,417 | 994 |

5 | 7,680 | 270,131 | 6,382 |

© 2012 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 license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Cantarella, J.; Cornish, J.; Mastin, M.; Parsley, J.
The 27 Possible Intrinsic Symmetry Groups of Two-Component Links. *Symmetry* **2012**, *4*, 129-142.
https://doi.org/10.3390/sym4010129

**AMA Style**

Cantarella J, Cornish J, Mastin M, Parsley J.
The 27 Possible Intrinsic Symmetry Groups of Two-Component Links. *Symmetry*. 2012; 4(1):129-142.
https://doi.org/10.3390/sym4010129

**Chicago/Turabian Style**

Cantarella, Jason, James Cornish, Matt Mastin, and Jason Parsley.
2012. "The 27 Possible Intrinsic Symmetry Groups of Two-Component Links" *Symmetry* 4, no. 1: 129-142.
https://doi.org/10.3390/sym4010129