Distinguishing Links - Part V

## APPLYING COLORIZATION INVARIANTS - Permutation Group S(6)

Once the 5-permutation invariants have been used, 82 out of the 118 links have been shown distinct. The remaining 36 links have been divided into the following 13 classes.

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

Colorization Invariant 3-1-1-1

The only class that is affected is

Class 5

• 28 ... 8-12 ... DISTINCT
• 10 ... 8-27 ... DISTINCT

None of the other classes splits. The number of links shown distinct has thus been increased to 84. The remaining 34 links belong into these 12 classes.

1. 3-1, 9-42, 9-66
2. 5-1, 8-18, 9-32
3. 6-3, 8-20, 9-28
4. 7-4, 7-10, 8-1, 8-17, 9-14, 9-58
5. 8-23, 9-34
6. 9-1, 9-3, 9-25
7. 9-11, 9-12, 9-16
8. 9-26, 9-39, 9-60
9. 9-29, 9-44
10. 9-38, 9-61
11. 9-50, 9-64
12. 9-55, 9-56

Colorization Invariant 3-2-1

Only the following classes are affected.

Class 3

• 13 ... 9-28 ... DISTINCT
• 7 ... 6-3, 8-20
Class 4
• 13 ... 8-1 ... DISTINCT
• 1 ... 7-4, 7-10, 8-17, 9-14, 9-58

None of the other classes splits. The number of links shown distinct has thus been increased to 86. The remaining 32 links belong into these 12 classes.

1. 3-1, 9-42, 9-66
2. 5-1, 8-18, 9-32
3. 6-3, 8-20
4. 7-4, 7-10, 8-17, 9-14, 9-58
5. 8-23, 9-34
6. 9-1, 9-3, 9-25
7. 9-11, 9-12, 9-16
8. 9-26, 9-39, 9-60
9. 9-29, 9-44
10. 9-38, 9-61
11. 9-50, 9-64
12. 9-55, 9-56

Colorization Invariant 4-1-1

The only class affected is

Class 12

• 57 ... 9-55 ... DISTINCT
• 41 ... 9-56 ... DISTINCT

None of the other classes splits. The number of links shown distinct has thus been increased to 88. The remaining 30 links belong into these 11 classes.

1. 3-1, 9-42, 9-66
2. 5-1, 8-18, 9-32
3. 6-3, 8-20
4. 7-4, 7-10, 8-17, 9-14, 9-58
5. 8-23, 9-34
6. 9-1, 9-3, 9-25
7. 9-11, 9-12, 9-16
8. 9-26, 9-39, 9-60
9. 9-29, 9-44
10. 9-38, 9-61
11. 9-50, 9-64

Colorization Invariant 4-2

Class 1

• 25 ... 9-42 ... DISTINCT
• 9 ... 3-1, 9-66
Class 2
• 33 ... 5-1 ... DISTINCT
• 17 ... 8-18, 9-32
Class 3
• 49 ... 8-20 ... DISTINCT
• 1 ... 6-3 ... DISTINCT
Class 4
• 17 ... 7-4 ... DISTINCT
• 1 ... 7-10, 8-17, 9-14, 9-58
Class 5
• 17 ... 8-23 ... DISTINCT
• 1 ... 9-34 ... DISTINCT
Class 6
• 17 ... 9-3 ... DISTINCT
• 1 ... 9-1, 9-25
Class 7
• 17 ... 9-11, 9-12
• 1 ... 9-16 ... DISTINCT
Class 8
• 25 ... 9-39 ... DISTINCT
• 9 ... 9-26, 9-60

Class 9 and Class 10 do not split.

Class 11

• 41 ... 9-50 ... DISTINCT
• 9 ... 9-64 ... DISTINCT

The number of links shown distinct has thus been increased to 100. The remaining 18 links belong into these 8 classes.

1. 3-1, 9-66
2. 7-10, 8-17, 9-14, 9-58
3. 8-18, 9-32
4. 9-1, 9-25
5. 9-11, 9-12
6. 9-26, 9-60
7. 9-29, 9-44
8. 9-38, 9-61

Colorization Invariant 5-1

Class 1 and Class 2 do not split.

Class 3

• 21 ... 8-18 ... DISTINCT
• 11 ... 9-32 ... DISTINCT

Class 4 does not split.

Class 5

• 21 ... 9-11 ... DISTINCT
• 1 ... 9-12 ... DISTINCT

Class 6 and Class 7 do not split.

Class 8

• 21 ... 9-61 ... DISTINCT
• 11 ... 9-38 ... DISTINCT

Therefore, once the 6-permutation invariants have been applied, 106 out of the original 118 links have been shown distinct, while the 12 remaining links belong to 5 classes. We shall continue the discussion in this page.

Charilaos Aneziris