## APPLYING COLORIZATION INVARIANTS - Permutation Group S(5)

We have already divided the 118 links into 33 classes. Out of these 33 classes, 21 contain just one link: 3-2, 7-2, 7-7, 8-2, 8-10, 8-11, 8-16, 8-24, 8-25, 8-26, 9-6, 9-8, 9-10, 9-19, 9-22, 9-43, 9-45, 9-49, 9-51, 9-52 and 9-62. These links have thus been shown distinct, and one need not calculate any more of their other invariants. The other 97 links are divided into 12 classes as follows.

1. 3-1, 8-21, 9-33, 9-42, 9-55, 9-56, 9-66
2. 4-1, 7-6, 7-8, 8-19, 8-23, 8-31, 9-1, 9-2, 9-3, 9-4, 9-11, 9-12, 9-16, 9-25, 9-34, 9-40, 9-41, 9-53, 9-54, 9-57, 9-59
3. 5-1, 5-3, 6-1, 6-3, 7-3, 7-4, 7-10, 8-1, 8-15, 8-17, 8-18, 8-20, 8-22, 8-28, 8-29, 8-30, 8-32, 9-14, 9-20, 9-24, 9-28, 9-29, 9-30, 9-32, 9-38, 9-44, 9-47, 9-58, 9-61, 9-63
4. 5-2, 9-15
5. 6-2, 8-8, 8-14, 9-5, 9-7, 9-9
6. 6-4, 7-1, 7-9, 9-26, 9-31, 9-39, 9-48, 9-50, 9-60, 9-64
7. 7-5, 9-27
8. 8-3, 9-21
9. 8-4, 8-5, 8-6, 8-12, 8-13, 8-27, 9-18, 9-23, 9-46
10. 8-7, 8-9, 9-13, 9-17
11. 9-35, 9-65
12. 9-36, 9-37
We are now going to apply the invariants obtained from the 5-permutation group.

Colorization Invariant 2-1-1-1

Class 5

• 10 ... 8-8, 8-14, 9-9
• 4 ... 6-2, 9-5, 9-7
Class 10
• 10 ... 8-7, 8-9, 9-13
• 4 ... 9-17 ... DISTINCT

None of the other ten classes is split into subclasses. We thus end up with thirteen new classes that contain two or more links, while the number of links shown distinct has been increased to 22. The new classes are related to the old ones as follows. The old class 5 splits into the new classes 5 (led by 6-2) and 11 (led by 8-8); the old class 10 becomes the new class 10 but without 9-17 that was shown distinct; the old classes 1, 2, 3, 4, 6, 7, 8, 9, 11, 12 now become the new classes 1, 2, 3, 4, 6, 7, 8, 9, 12, 13 respectively.

Colorization Invariant 2-2-1

Class 1

• 13 ... 9-33 ... DISTINCT
• 5 ... 3-1, 8-21, 9-42, 9-55, 9-56, 9-66
Class 2
• 49 ... 9-54 ... DISTINCT
• 9 ... 4-1, 9-1, 9-2, 9-3, 9-4, 9-11, 9-12, 9-16, 9-25, 9-41, 9-57
• 1 ... 7-6, 7-8, 8-19, 8-23, 8-31, 9-34, 9-40, 9-53, 9-59
Class 3
• 9 ... 5-3, 8-15, 8-29, 8-30, 9-30
• 1 ... 5-1, 6-1, 6-3, 7-3, 7-4, 7-10, 8-1, 8-17, 8-18, 8-20, 8-22, 8-28, 8-32, 9-14, 9-20, 9-24, 9-28, 9-29, 9-32, 9-38, 9-44, 9-47, 9-58, 9-61, 9-63
Class 4
• 11 ... 9-15 ... DISTINCT
• 3 ... 5-2 ... DISTINCT

Class 5 does not split.

Class 6

• 13 ... 7-9, 9-48
• 5 ... 6-4, 7-1, 9-26, 9-31, 9-39, 9-50, 9-60, 9-64
Class 7
• 11 ... 7-5 ... DISTINCT
• 3 ... 9-27 ... DISTINCT Class 8
• 27 ... 9-21 ... DISTINCT
• 3 ... 8-3 ... DISTINCT Class 9
• 13 ... 8-6, 9-23
• 5 ... 8-4, 8-5, 8-12, 8-13, 8-27, 9-18, 9-46
Class 10
• 11 ... 8-7 ... DISTINCT
• 3 ... 8-9, 9-13

Class 11 does not split.

Class 12

• 25 ... 9-35 ... DISTINCT
• 17 ... 9-65 ... DISTINCT

Class 13 does not splt.

We end up with 33 classes containing just one link each, and 13 classes containing the other 85 links. These 13 new classes consist of the following links.

1. 3-1, 8-21, 9-42, 9-55, 9-56, 9-66
2. 4-1, 9-1, 9-2, 9-3, 9-4, 9-11, 9-12, 9-16, 9-25, 9-41, 9-57
3. 5-1, 6-1, 6-3, 7-3, 7-4, 7-10, 8-1, 8-17, 8-18, 8-20, 8-22, 8-28, 8-32, 9-14, 9-20, 9-24, 9-28, 9-29, 9-32, 9-38, 9-44, 9-47, 9-58, 9-61, 9-63
4. 5-3, 8-15, 8-29, 8-30, 9-30
5. 6-2, 9-5, 9-7
6. 6-4, 7-1, 9-26, 9-31, 9-39, 9-50, 9-60, 9-64
7. 7-6, 7-8, 8-19, 8-23, 8-31, 9-34, 9-40, 9-53, 9-59
8. 7-9, 9-48
9. 8-4, 8-5, 8-12, 8-13, 8-27, 9-18, 9-46
10. 8-6, 9-23
11. 8-8, 8-14, 9-9
12. 8-9, 9-13
13. 9-36, 9-37

Colorization Invariant 3-1-1

Class 1

• 13 ... 8-21 ... DISTINCT
• 7 ... 3-1, 9-42, 9-55, 9-56, 9-66
Class 2
• 13 ... 9-11, 9-12, 9-16
• 7 ... 4-1, 9-1, 9-2, 9-3, 9-4, 9-25, 9-41, 9-57
Class 3
• 7 ... 6-1, 8-22, 8-28, 9-24
• 1 ... 5-1, 6-3, 7-3, 7-4, 7-10, 8-1, 8-17, 8-18, 8-20, 8-32, 9-14, 9-20, 9-28, 9-29, 9-32, 9-38, 9-44, 9-47, 9-58, 9-61, 9-63
Class 4
• 7 ... 5-3, 8-29
• 1 ... 8-15, 8-30, 9-30

Class 5 does not split.

Class 6

• 7 ... 9-31 ... DISTINCT
• 1 ... 6-4, 7-1, 9-26, 9-39, 9-50, 9-60, 9-64

Class 7 does not split.

Class 8

• 7 ... 7-9 ... DISTINCT
• 1 ... 9-48 ... DISTINCT
Class 9
• 19 ... 9-46 ... DISTINCT
• 13 ... 8-5 ... DISTINCT
• 7 ... 8-4, 8-12, 8-13, 8-27, 9-18
Class 10
• 19 ... 8-6 ... DISTINCT
• 13 ... 9-23 ... DISTINCT
Class 11
• 8 ... 8-14 ... DISTINCT
• 2 ... 8-8, 9-9

Class 12 does not split.

Class 13

• 13 ... 9-36 ... DISTINCT
• 7 ... 9-37 ... DISTINCT
We thus reach a point where there are 44 classes with just one link each, and 13 classes containing the remaining 74 links. These are numbered as follows.
1. 3-1, 9-42, 9-55, 9-56, 9-66
2. 4-1, 9-1, 9-2, 9-3, 9-4, 9-25, 9-41, 9-57
3. 5-1, 6-3, 7-3, 7-4, 7-10, 8-1, 8-17, 8-18, 8-20, 8-32, 9-14, 9-20, 9-28, 9-29, 9-32, 9-38, 9-44, 9-47, 9-58, 9-61, 9-63
4. 5-3, 8-29
5. 6-1, 8-22, 8-28, 9-24
6. 6-2, 9-5, 9-7
7. 6-4, 7-1, 9-26, 9-39, 9-50, 9-60, 9-64
8. 7-6, 7-8, 8-19, 8-23, 8-31, 9-34, 9-40, 9-53, 9-59
9. 8-4, 8-12, 8-13, 8-27, 9-18
10. 8-8, 9-9
11. 8-9, 9-13
12. 8-15, 8-30, 9-30
13. 9-11, 9-12, 9-16

Colorization Invariant 3-2

Class 1

• 7 ... 9-55, 9-56
• 1 ... 3-1, 9-42, 9-66
Class 2
• 19 ... 9-57 ... DISTINCT
• 13 ... 4-1 ... DISTINCT
• 7 ... 9-2, 9-4, 9-41
• 1 ... 9-1, 9-3, 9-25
Class 3
• 13 ... 9-63 ... DISTINCT
• 7 ... 5-1, 6-3, 8-18, 8-20, 9-28, 9-32
• 1 ... 7-3, 7-4, 7-10, 8-1, 8-17, 8-32, 9-14, 9-20, 9-29, 9-38, 9-44, 9-47, 9-58, 9-61

Class 4 does not split.

Class 5

• 7 ... 8-22, 9-24
• 1 ... 6-1, 8-28
Class 6
• 8 ... 9-7 ... DISTINCT
• 2 ... 6-2, 9-5
Class 7
• 7 ... 7-1, 9-26, 9-39, 9-60
• 1 ... 6-4, 9-50, 9-64
Class 8
• 7 ... 7-8, 9-59
• 1 ... 7-6, 8-19, 8-23, 8-31, 9-34, 9-40, 9-53
Class 9
• 13 ... 8-13 ... DISTINCT
• 1 ... 8-4, 8-12, 8-27, 9-18

Class 10 does not split.

Class 11 does not split.

Class 12

• 7 ... 8-15, 9-30
• 1 ... 8-30 ... DISTINCT

Class 13 does not split.

We thus reached a point with 50 classes containing just one link each, while the following 19 classes contain the other 68 links.

1. 3-1, 9-42, 9-66
2. 5-1, 6-3, 8-18, 8-20, 9-28, 9-32
3. 5-3, 8-29
4. 6-1, 8-28
5. 6-2, 9-5
6. 6-4, 9-50, 9-64
7. 7-1, 9-26, 9-39, 9-60
8. 7-3, 7-4, 7-10, 8-1, 8-17, 8-32, 9-14, 9-20, 9-29, 9-38, 9-44, 9-47, 9-58, 9-61
9. 7-6, 8-19, 8-23, 8-31, 9-34, 9-40, 9-53
10. 7-8, 9-59
11. 8-4, 8-12, 8-27, 9-18
12. 8-8, 9-9
13. 8-9, 9-13
14. 8-15, 9-30
15. 8-22, 9-24
16. 9-1, 9-3, 9-25
17. 9-2, 9-4, 9-41
18. 9-11, 9-12, 9-16
19. 9-55, 9-56

Colorization Invariant 4-1

Class 1 and Class 2 do not split.

Class 3

• 5 ... 5-3 ... DISTINCT
• 1 ... 8-29 ... DISTINCT
Class 4
• 9 ... 8-28 ... DISTINCT
• 1 ... 6-1 ... DISTINCT

Class 5 does not split.

Class 6

• 13 ... 6-4 ... DISTINCT
• 9 ... 9-50, 9-64
Class 7
• 9 ... 7-1 ... DISTINCT
• 5 ... 9-26, 9-39, 9-60
Class 8
• 17 ... 9-20 ... DISTINCT
• 13 ... 7-3 ... DISTINCT
• 9 ... 9-29, 9-44
• 5 ... 9-47 ... DISTINCT
• 1 ... 7-4, 7-10, 8-1, 8-17, 8-32, 9-14, 9-38, 9-58, 9-61
Class 9
• 9 ... 7-6, 9-53
• 5 ... 8-19, 8-31
• 1 ... 8-23, 9-34, 9-40

Class 10 does not split.

Class 11

• 21 ... 8-4, 9-18
• 13 ... 8-12, 8-27

Class 12 and Class 13 do not split.

Class 14

• 9 ... 8-15 ... DISTINCT
• 1 ... 9-30 ... DISTINCT

Class 15 and Class 16 do not split.

Class 17

• 17 ... 9-2 ... DISTINCT
• 13 ... 9-4 ... DISTINCT
• 1 ... 9-41 ... DISTINCT

Class 18 and Class 19 do not split.

We thus reached a point with 64 classes containing just one link each, and the following 15 classes containing the other 54 links.

1. 3-1, 9-42, 9-66
2. 5-1, 6-3, 8-18, 8-20, 9-28, 9-32
3. 6-2, 9-5
4. 7-4, 7-10, 8-1, 8-17, 8-32, 9-14, 9-38, 9-58, 9-61
5. 7-6, 9-53
6. 7-8, 9-59
7. 8-4, 9-18
8. 8-8, 9-9
9. 8-9, 9-13
10. 8-12, 8-27
11. 8-19, 8-31
12. 8-22, 9-24
13. 8-23, 9-34, 9-40
14. 9-1, 9-3, 9-25
15. 9-11, 9-12, 9-16
16. 9-26, 9-39, 9-60
17. 9-29, 9-44
18. 9-50, 9-64
19. 9-55, 9-56

Colorization Invariant 5

Class 1 does not split.

Class 2

• 6 ... 5-1, 8-18, 9-32
• 1 ... 6-3, 8-20, 9-28
Class 3
• 19 ... 9-5 ... DISTINCT
• 14 ... 6-2 ... DISTINCT
Class 4
• 11 ... 8-32 ... DISTINCT
• 6 ... 9-38, 9-61
• 1 ... 7-4, 7-10, 8-1, 8-17, 9-14, 9-58
Class 5
• 11 ... 9-53 ... DISTINCT
• 1 ... 7-6 ... DISTINCT
Class 6
• 6 ... 9-59 ... DISTINCT
• 1 ... 7-8 ... DISTINCT
Class 7
• 11 ... 8-4 ... DISTINCT
• 6 ... 9-18 ... DISTINCT
Class 8
• 14 ... 9-9 ... DISTINCT
• 9 ... 8-8 ... DISTINCT
Class 9
• 14 ... 9-13 ... DISTINCT
• 9 ... 8-9 ... DISTINCT

Class 10 does not split.

Class 11

• 6 ... 8-31 ... DISTINCT
• 1 ... 8-19 ... DISTINCT
Class 12
• 6 ... 8-22 ... DISTINCT
• 1 ... 9-24 ... DISTINCT
Class 13
• 6 ... 8-23, 9-34
• 1 ... 9-40 ... DISTINCT

The remaining classes (14, 15, 16, 17, 18 and 19) do not split.

Thus, once all 5-permutation colorizations have been applied, we obtain 82 classes containing just one link each, and 13 classes containing the remaining 36 links. We continue the discussion with the 6-permutation colorizations in this page.

Charilaos Aneziris