## APPLYING COLORIZATION INVARIANTS - Permutation Group S(4)

We have already divided the 118 links into 17 classes. Out of these 17 classes, 6 contain just one link: 7-7, 8-10, 8-25, 8-26, 9-45 and 9-52. These links have thus been shown distinct, and one need not calculate any of their other invariants. The other 112 links are divided into 11 classes as follows.

1. 3-1, 6-4, 7-1, 7-9, 8-4, 8-5, 8-6, 8-12, 8-13, 8-21, 8-27, 9-18, 9-23, 9-26, 9-31, 9-33, 9-39, 9-42, 9-46, 9-48, 9-50, 9-55, 9-56, 9-60, 9-62, 9-64, 9-66
2. 3-2, 8-16, 9-8
3. 4-1, 5-1, 5-3, 6-1, 6-3, 7-3, 7-4, 7-6, 7-8, 7-10, 8-1, 8-15, 8-17, 8-18, 8-19, 8-20, 8-22, 8-23, 8-28, 8-29, 8-30, 8-31, 8-32, 9-1, 9-2, 9-3, 9-4, 9-11, 9-12, 9-14, 9-16, 9-20, 9-24, 9-25, 9-28, 9-29, 9-30, 9-32, 9-34, 9-38, 9-40, 9-41, 9-44, 9-47, 9-53, 9-54, 9-57, 9-58, 9-59, 9-61, 9-63
4. 5-2, 8-7, 8-9, 9-13, 9-15, 9-17
5. 6-2, 8-3, 8-8, 8-14, 9-5, 9-7, 9-9, 9-21
6. 7-2, 9-6
7. 7-5, 9-22, 9-27
8. 8-2, 9-19
9. 8-11, 9-10
10. 8-24, 9-35, 9-36, 9-37, 9-51, 9-65
11. 9-43, 9-49
We are now going to apply the invariants obtained from the 4-permutation group.

Colorization Invariant 2-1-1

Class 1

• 9 ... 8-4, 8-5, 8-6, 8-12, 8-13, 8-27, 9-18, 9-23, 9-46, 9-62
• 5 ... 3-1, 6-4, 7-1, 7-9, 8-21, 9-26, 9-31, 9-33, 9-39, 9-42, 9-48, 9-50, 9-55, 9-56, 9-60, 9-64, 9-66
Class 10
• 21 ... 9-35, 9-36, 9-37, 9-65
• 17 ... 8-24, 9-51

None of the other nine classes is split into subclasses, because all their links have equal invariants. We thus end up with thirteen classes: the old class 1 is split into the new classes 1 (led by 3-1) and 9 (led by 8-4); the old class 10 is split into the new classes 11 (led by 8-24) and 12 (led by 9-35); and the old classes 2, 3, 4, 5, 6, 7, 8, 9, 11 become the new classes 2, 3, 4, 5, 6, 7, 8, 10, 13 respectively.

Colorization Invariant 3-1

Class 1

• 4 ... 3-1, 8-21, 9-33, 9-42, 9-55, 9-56, 9-66
• 1 ... 6-4, 7-1, 7-9, 9-26, 9-31, 9-39, 9-48, 9-50, 9-60, 9-64
Class 2
• 8 ... 9-8 ... DISTINCT
• 5 ... 8-16 ... DISTINCT
• 2 ... 3-2 ... DISTINCT
Class 3
• 4 ... 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
• 1 ... 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
Class 4
• 5 ... 5-2, 9-15
• 2 ... 8-7, 8-9, 9-13, 9-17
Class 5
• 8 ... 8-3, 9-21
• 2 ... 6-2, 8-8, 8-14, 9-5, 9-7, 9-9
Class 6
• 5 ... 7-2 ... DISTINCT
• 2 ... 9-6 ... DISTINCT
Class 7
• 5 ... 9-22 ... DISTINCT
• 2 ... 7-5, 9-27
Class 8
• 8 ... 9-19 ... DISTINCT
• 5 ... 8-2 ... DISTINCT
Class 9
• 16 ... 9-62 ... DISTINCT
• 4 ... 8-4, 8-5, 8-6, 8-12, 8-13, 8-27, 9-18, 9-23, 9-46
Class 10
• 13 ... 9-10 ... DISTINCT
• 7 ... 8-11 ... DISTINCT
Class 11
• 16 ... 8-24 ... DISTINCT
• 1 ... 9-51 ... DISTINCT
Class 12
• 4 ... 9-35, 9-65
• 1 ... 9-36 9-37
Class 13
• 5 ... 9-49 ... DISTINCT
• 2 ... 9-43 ... DISTINCT

At this point the 118 links have been split into 33 classes; 21 classes consist of exactly 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, 9-62) while the other 12 classes contain the remaining 97 links. We will continue the discussion with the invariants of the S(5) group in this page.

Charilaos Aneziris