## APPLYING COLORIZATION INVARIANTS - Permutation Group S(3)

After applying the "component number" and the "linking number" invariants, we showed that the 118 links of our list can be divided into the following eight classes:

1. 3-1, 4-1, 5-1, 5-3, 6-1, 6-3, 6-4, 7-1, 7-3, 7-4, 7-6, 7-8, 7-9, 7-10, 8-1, 8-4, 8-5, 8-6, 8-12, 8-13, 8-15, 8-17, 8-18, 8-19, 8-20, 8-21, 8-22, 8-23, 8-24, 8-27, 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, 9-18, 9-20, 9-23, 9-24, 9-25, 9-26, 9-28, 9-29, 9-30, 9-31, 9-32, 9-33, 9-34, 9-35, 9-36, 9-37, 9-38, 9-39, 9-40, 9-41, 9-42, 9-44, 9-46, 9-47, 9-48, 9-50, 9-51, 9-53, 9-54, 9-55, 9-56, 9-57, 9-58, 9-59, 9-60, 9-61, 9-62, 9-63, 9-64, 9-65, 9-66
2. 3-2, 8-2, 8-16, 9-8, 9-19
3. 5-2, 7-2, 8-7, 8-9, 9-6, 9-13, 9-15, 9-17, 9-52
4. 6-2, 8-3, 8-8, 8-14, 8-25, 9-5, 9-7, 9-9, 9-21
5. 7-5, 7-7, 9-22, 9-27
6. 8-10, 8-11, 9-10
7. 8-26
8. 9-43, 9-45, 9-49

Any link belonging to one class, has been shown distinct to links belonging to any other class, since their component numbers and/or linking numbers are different. We shall now show how the colorization invariants split these links to even more (and smaller) classes, until they are all shown distinct from each other.

Notation: each invariant is indicated through the "conjugacy class of the permutation group" to which it belongs. Therefore the invariant 2-1 denotes colorizations where the acceptable "colors" are the permutations (1->2,2->1,3->3), (1->3,2->2,3->1) and (1->1,2->3,3->2). Each link will be associated to a number that indicates the number of acceptable colorizations, once the "color" of one strand is fixed. Therefore if 2-1 maps a link to the number 1, this means that if one strand of the link is mapped to the "color" (1->2,2->1,3->3), then the only acceptable colorization is the one where all strands are mapped to the same "color".

Colorization Invariant 2-1

Class 1

• 9 ... 8-24, 9-35, 9-36, 9-37, 9-51, 9-65
• 3 ... 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
• 1 ... 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
Class 2
• 3 ... 8-2, 9-19
• 1 ... 3-2, 8-16, 9-8
Class 3
• 9 ... 9-52 ... DISTINCT
• 3 ... 7-2, 9-6
• 1 ... 5-2, 8-7, 8-9, 9-13, 9-15, 9-17
Class 4
• 3 ... 8-25 ... DISTINCT
• 1 ... 6-2, 8-3, 8-8, 8-14, 9-5, 9-7, 9-9, 9-21
Class 5
• 3 ... 7-7 ... DISTINCT
• 1 ... 7-5, 9-22, 9-27
Class 6
• 3 ... 8-10 ... DISTINCT
• 1 ... 8-11, 9-10

Class 8

• 3 ... 9-45 ... DISTINCT
• 1 ... 9-43, 9-49

Thus, after applying the 2-1 invariant, the original 8 classes split into 17. Out of these 17 new classes, 6 contain just one link, and thus these links have been proven to be distinct. We continue in this page with further colorization invariants.

Charilaos Aneziris