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

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

Class 4

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

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

Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8

Class 9 and Class 10 do not split.

Class 11

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

Class 4 does not split.

Class 5

Class 6 and Class 7 do not split.

Class 8

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

Copyright 2002

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