# Intrinsic Symmetry Groups of Links with 8 and Fewer Crossings

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

`SnapPea`method gives much insight into what those isotopies might look like. In addition, it is worth noting that

`SnapPea`is a large and complicated computer program, and while its results are accurate for the links in our table, it is always worthwhile to have alternate proofs for results that depend essentially on nontrivial computer calculations.

## 2. The Symmetry and Intrinsic Symmetry Groups

## 3. Methods and Notation

`SnapPea`. The component numbers and orientations in

`SnapPea`seem to have been chosen arbitrarily sometime in the $1980s$ by Joe Christy when he digitized the Rolfsen table [16].

`SnapPea`calculations, we used the Python interface provided by

`SnapPy`[17] to compute the image of $\pi :MCG({S}^{3},L)\to MCG\left({S}^{3}\right)\times MCG\left(L\right)$. The results agreed [18] with the tables we give below, meaning that our results serve as an independent verification of

`SnapPea`’s accuracy for all of these links except ${8}_{5}^{3}$ (which uses SnapPea to rule out a single potential symmetry).

## 4. The Whitten Group

**Definition**

**4.1**

#### 4.1. Link Operations

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

**Example**

**4.2**

**Example**

**4.3**

**Proposition**

**4.4**

**Definition**

**4.5**

**Example**

**4.6**

**Example**

**4.7**

**Example**

**4.8**

**Proposition**

**4.9**

## 5. The Linking Matrix

**Definition**

**5.1**

**Definition**

**5.2**

**Proposition**

**5.3**

**Corollary**

**5.4**

#### 5.1. Linking Matrix for Three-Component Links

**Proposition**

**5.5**

**Lemma**

**5.6**

**Example**

**5.7**

#### 5.2. Linking Matrix for Four-Component Links

**Proposition**

**5.8**

**Lemma**

**5.9**

**Corollary**

**5.10**

## 6. The Satellite Lemma

**Definition**

**6.1**

**Definition**

**6.2**

**Lemma**

**6.3**

## 7. Two-Component Links

#### 7.1. Symmetry Names and Results

**Question**

**7.1**

**Theorem**

**7.2**

#### 7.2. Proofs for Two-Component Links

**Lemma**

**7.3**

**Lemma**

**7.4**

**Lemma**

**7.5**

**Lemma**

**7.6.**

- 1.
- If the linking number $Lk\left(L\right)\ne 0$, then $\Sigma \left(L\right)<{\Sigma}_{8,2}$.
- 2.
- For L alternating, if the self-writhe $s\left(L\right)\ne 0$, then $\Sigma \left(L\right)<{\Sigma}_{8,1}$.
- 3.
- For L alternating, if $Lk\left(L\right)\ne 0$ and $s\left(L\right)\ne 0$, then $\Sigma \left(L\right)<{\Sigma}_{4,1}$.

**Lemma**

**7.7**

- 1.
- If L is purely invertible, then ${\Sigma}_{2,1}<\Sigma \left(L\right)$.
- 2.
- If the components of L are different knot types, then $\Sigma \left(L\right)<{\Sigma}_{8,3}$.
- 3.
- If both hypotheses above are true, and
- (a)
- if $lk\left(L\right)\ne 0$, then $\Sigma \left(L\right)$ is either ${\Sigma}_{2,1}$ or ${\Sigma}_{4,3}$.
- (b)
- if L is alternating and $s\left(L\right)\ne 0$, then $\Sigma \left(L\right)$ is either ${\Sigma}_{2,1}$ or ${\Sigma}_{4,2}$.

#### 7.2.1. Links with Symmetry Group ${\Sigma}_{2,1}$

**Claim**

**7.8**

#### 7.2.2. Links with Symmetry Group ${\Sigma}_{4,1}$

**Claim**

**7.9**

**Claim**

**7.10**

#### 7.2.3. Links with Symmetry Group ${\Sigma}_{4,2}$

**Claim**

**7.11**

**Claim**

**7.12**

**Claim**

**7.13**

#### 7.2.4. Links with Symmetry Group ${\Sigma}_{8,1}$

**Claim**

**7.14**

#### 7.2.5. Links with Symmetry Group ${\Sigma}_{8,2}$

**Claim**

**7.15**

## 8. Three-Component Links

- PI, for pure invertibility, i.e., the element $(1,-1,-1,-1,e)$
- PE, for having all pure exchanges, i.e., all elements $(1,1,1,1,p)$ where $p\in {S}_{3}$

**Claim**

**8.1**

**Claim**

**8.2**

- (a)
- ${\u03f5}_{1}{\u03f5}_{2}{\u03f5}_{3}=1$ and p is an even permutation, or
- (b)
- ${\u03f5}_{1}{\u03f5}_{2}{\u03f5}_{3}=-1$ and p is odd.

**Claim**

**8.3**

**Claim**

**8.4**

**Claim**

**8.5**

**Claim**

**8.6**

**Claim**

**8.7**

**Claim**

**8.8**

**Claim**

**8.9**

**Claim**

**8.10**

**Claim**

**8.11**

**Claim**

**8.12**

**Claim**

**8.13**

**Claim**

**8.14**

## 9. Isotopies for Four-Component Links

**Claim**

**9.1**

**Claim**

**9.2**

**Claim**

**9.3**

## 10. Comparison of Intrinsic Symmetry Groups with Ordinary Symmetry Groups for Links

**Lemma**

**10.1**

**Lemma**

**10.2**

## 11. Future Directions

`KnotTheory`to systematically apply all possible Whitten group elements to each link and then check the knot types of the components, the linking matrix, the Jones polynomial, and the HOMFLYPT polynomial in an attempt to distinguish the new link from the original one. We then checked the computer calculations by hand. This automated method clearly cannot compute $\Sigma \left(L\right)$, but it does provide a subgroup ${\Sigma}^{\prime}\left(L\right)$ of $\Gamma \left(L\right)$ which is known to contain $\Sigma \left(L\right)$. While we do not currently intend to generate isotopies for links with higher crossing number, we intend to present our computationally-obtained ${\Sigma}^{\prime}\left(L\right)$ groups for 9, 10, and 11 crossing links in a future publication.

## Acknowledgements

## A. Guide to Link Isotopy Figures

- Appendix A explains the moves depicted in these isotopies.
- Appendix B exhibits 41 isotopies which can be found by simply rotating about one axis.
- Appendix C contains the remaining isotopies for 2-component links
- Appendix D contains the remaining isotopies for 3-component links
- Appendix E contains the remaining isotopies for 4-component links

**Figure A2.**The left figure shows a “flip” move. In this move, we take the portion of the link in the dotted box and flip it in the direction indicated by the curved arrows, resulting in the diagram shown on the right. Arrows denote transitions between stages of the isotopy. The right figure shows a simpler transformation. Here the arrow shows a portion of the link. Whether this portion moves over or under intervening portions of the link should be clear from the next drawing.Examples of moves in isotopy diagrams.

## B. Isotopy Figures Found by Rotations

#### B.1. Pure Exchange Isotopies Found by Rotation

**Figure B2.**Pure exchange isotopies found by rotation about the z-axis. In the case of links with more than two components, all possible exchanges are listed underneath the respective figures.

#### B.2. Pure Invertibility Isotopies Found by Rotation

#### B.3. Other Isotopies Found by Rotation

## C. Isotopy Figures for Two-Component Links

**Figure C1.**${\left({2}_{1}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(-1,1,-1,e)$.

**Figure C2.**${\left({6}_{2}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(-1,1,-1,e)$.

**Figure C3.**${\left({7}_{4}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,1,e)$.

**Figure C4.**${\left({7}_{6}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,1,e)$.

**Figure C5.**${\left({8}_{8}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(-1,1,-1,e)$.

**Figure C6.**${\left({8}_{10}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,1,e)$.

**Figure C7.**${\left({8}_{12}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,1,e)$.

**Figure C8.**${\left({8}_{13}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,1,e)$.

**Figure C9.**${\left({8}_{15}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,-1,e)$.

#### C.1. Isotopies Showing Pure Exchange Symmetries for Two-Component Links

**Figure C10.**${\left({7}_{1}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,\left(12\right))$.

**Figure C11.**${\left({7}_{2}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,\left(12\right))$.

**Figure C12.**${\left({8}_{2}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,\left(12\right))$.

**Figure C13.**${\left({8}_{3}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,\left(12\right))$.

**Figure C14.**${\left({8}_{7}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,\left(12\right))$.

**Figure C15.**${\left({8}_{8}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,\left(12\right))$.

**Figure C16.**The links ${5}_{1}^{2}$, ${6}_{3}^{2}$, ${7}_{3}^{2}$ and ${8}_{6}^{2}$ share the common form above, with 1, 2, 3, and 4 crossings replacing the dots in the central “twisted” region of component 2 above. The pure exchange symmetry for each can be accomplished in a similar way, by shifting twists from component 2 to component 1 as shown. Since the orientations vary between links, we do not show arrows above. Some examples require a final flip or twist to match orientations, but in each case it is not difficult to figure out the required move.Pure exchange isotopies for ${5}_{1}^{2}$, ${6}_{3}^{2}$, ${7}_{3}^{2}$ and ${8}_{6}^{2}$.

#### C.2. Isotopies Showing Pure Invertibility for Two-Component Links

**Figure C17.**${\left({6}_{2}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,-1,e)$.

**Figure C18.**${\left({7}_{2}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,-1,e)$.

**Figure C19.**${\left({8}_{2}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,-1,e)$.

**Figure C20.**${\left({8}_{5}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,-1,e)$.

**Figure C21.**${\left({8}_{8}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,-1,e)$.

**Figure C22.**${\left({8}_{10}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,-1,e)$.

**Figure C23.**${\left({8}_{12}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,-1,e)$.

**Figure C24.**${\left({8}_{15}^{2}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,-1,e)$.

## D. Isotopy Figures for Three-Component Links

**Figure D1.**${\left({6}_{1}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,-1,-1,e)$.

**Figure D2.**${\left({6}_{1}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,1,\left(23\right))$.

**Figure D3.**${\left({6}_{2}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(-1,-1,1,1,\left(13\right))$.

**Figure D4.**${\left({6}_{2}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(-1,1,1,1,e)$.

**Figure D5.**${\left({6}_{3}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,-1,1,\left(132\right))$.

**Figure D6.**${\left({6}_{3}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,-1,-1,\left(12\right))$.

**Figure D7.**${\left({7}_{1}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,-1,1,\left(123\right))$.

**Figure D8.**${\left({7}_{1}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,1,\left(23\right))$.

**Figure D9.**${\left({8}_{1}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,1,1,\left(23\right))$.

**Figure D10.**${\left({8}_{2}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,1,\left(23\right))$.

**Figure D11.**${\left({8}_{3}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,-1,-1,e)$ .

**Figure D12.**${\left({8}_{3}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,1,\left(12\right))$.

**Figure D13.**${\left({8}_{3}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,1,\left(13\right))$.

**Figure D14.**${\left({8}_{4}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(-1,1,1,1,\left(12\right))$.

**Figure D15.**${\left({8}_{5}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,1,1,\left(23\right))$.

**Figure D16.**${\left({8}_{5}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,-1,-1,e)$.

**Figure D17.**${\left({8}_{6}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,1,1,\left(23\right))$.

**Figure D18.**${\left({8}_{6}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(-1,1,-1,-1,e)$.

**Figure D19.**${\left({8}_{7}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,1,\left(23\right))$.

**Figure D20.**${\left({8}_{8}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,-1,1,1,\left(23\right))$.

**Figure D21.**${\left({8}_{8}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,-1,-1,\left(23\right))$.

**Figure D22.**${\left({8}_{9}^{3}\right)}^{\gamma},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\gamma =(1,1,1,1,\left(23\right))$.