The lists below are presented in a plain text format as two matrices one for the up action (U) and one for the down action (D). Each matrix is specified row by row. All of the lists are presented as biracks, that is with both U and D specified; for the quandles and racks, the D action is the identity.

• 3 biracks of size 2 (includes quandles, biquandles and racks)
• 2 biquandle of size 2 (includes quandles)
• 2 racks of size 2 (includes quandles)
• 1 quandle of size 2
• There are no biracks of size 2 that are not biquandles and not racks
• 1 biquandle of size 2 that is not a quandle
• 1 rack of size 2 that is not a quandle
• 2 quandle-related biracks of size 2
• 1 quandle-related birack of size 2 that is not a quandle
• There are no quandle-related biquandles of size 2 that are not quandles
• There are no virtual pairs of size 2 that are not weld pairs
• 2 weld pairs of size 2
• There are no essential pairs of size 2


• 16 biracks of size 3 (includes quandles, biquandles and racks)
• 10 biquandles of size 3 (includes quandles)
• 6 racks of size 3 (includes quandles)
• 3 quandles of size 3
• 3 biracks of size 3 that are not biquandles and not racks
• 7 biquandles of size 3 that are not quandles
• 3 racks of size 3 that are not quandles
• 9 quandle-related biracks of size 3
• 6 quandle-related biracks of size 3 that are not quandles
• 1 quandle-related biquandle of size 3 that is not a quandle
• 5 virtual pairs of size 3 that are not weld pairs
• 16 weld pairs of size 3 that are not essential pairs
• 2 essential pairs of size 3


• 147 biracks of size 4 (includes quandles, biquandles and racks)
• 64 biquandle of size 4 (includes quandles)
• 19 racks of size 4 (includes quandles)
• 7 quandles of size 4
• 71 biracks of size 4 that are not biquandles and not racks
• 57 biquandles of size 4 that are not quandles
• 12 racks of size 4 that are not quandles
• 60 quandle-related biracks of size 4
• 53 quandle-related biracks of size 4 that are not quandles
• 11 quandle-related biquandles of size 4 that are not quandles
• 45 virtual pairs of size 4 that are not weld pairs
• 290 weld pairs of size 4 that are not essential pairs
• 8 essential pairs of size 4


• 703 quandle-related biracks of size 5 (includes quandles, biquandles and racks)
• 134 quandle-related biquandles of size 5 (includes quandles)
• 73 racks of size 5 (includes quandles)
• 21 quandles of size 5
• 517 quandle-related biracks of size 5 that are not biquandles and not racks
• 682 quandle-related biracks of size 5 that are not quandles (includes biquandles and racks)
• 113 quandle-related biquandles of size 5 that are not quandles
• 52 racks of size 5 that are not quandles
• 41 quandle-related virtual pairs of size 5 that are not weld pairs
• 454 quandle-related weld pairs of size 5 that are not essential pairs
• 17 quandle-related essential pairs of size 5


• 13562 quandle-related biracks of size 6 (includes quandles, biquandles and racks)
• 1578 quandle-related biquandles of size 6 (includes quandles)
• 352 racks of size 6 (includes quandles)
• 72 quandles of size 6
• 11704 quandle-related biracks of size 6 that are not biquandles and not racks
• 13490 quandle-related biracks of size 6 that are not quandles (includes biquandles and racks)
• 1506 quandle-related biquandles of size 6 that are not quandles
• 280 racks of size 6 that are not quandles
• 567 quandle-related virtual pairs of size 6 that are not weld pairs
• 9575 quandle-related weld pairs of size 6 that are not essential pairs
• 271 quandle-related essential pairs of size 6

The essential pairs above may be used by the braid programme to calculate invariants of welded knots. Using this approach a list of distinct non-trivial welded knots has been produced.

The lists were calculated using the programmes:

• nquandle
• nswitch
• biquandle-search
• dominant-pairs

Steps to calculate the above lists

The following steps were used to calculate the above lists for the cases k=2,3,4,

1. Calculate a list of all distinct biracks.

   biquandle-search -n=kb
   mv biquandle.out birack-k.txt

2. Determine the set of biquandles from the list of distinct biracks.

   biquandle-search -n=kR birack-k.txt
   mv biquandle.out biquandle-k.txt

3. Remove the biquandles from the list of distinct biracks. The biquandle search that created biquandle-k.txt only added a condition to filter out biracks from birack-k.txt, so is a strict subset of that set of biracks, therefore we do not need to consider permuataions when removing them.

   biquandle-search -n=kbRXex birack-k.txt biquandle-k.txt
   mv biquandle.out birack-k-not-biquandle.txt

4. Determine the set of racks from the list of biracks.

   biquandle-search -n=kr birack-k.txt
   mv biquandle.out rack-k.txt

5. Remove the racks from the list of biracks that are not biquandles.

   biquandle-search -n=kbRXe birack-k-not-biquandle.txt rack-k.txt
   mv biquandle.out birack-k-not-biquandle-not-rack-k.txt

6. Determine the quandles from the list of distinct biracks.

   biquandle-search -n=kq birack-k.txt
   mv biquandle.out quandle-k.txt

7. Remove the quandles from the list of biquandles.

   biquandle-search -n=kRXe biquandle-k.txt quandle-k.txt
   mv biquandle.out biquandle-k-not-quandle.txt

8. Remove the quandles from the list of racks.

   biquandle-search -n=krXe rack-k.txt quandle-k.txt
   mv biquandle.out rack-k-not-quandle.txt

9. Calculate the quandle-related biracks from the list of distinct biracks.

   biquandle-search -n=kbRQ birack-k.txt
   mv biquandle.out quandle-related-birack-k.txt

10. Remove the quandles from the quandle-related biracks.

    biquandle-search -n=kbRXe quandle-related-birack-k.txt quandle-k.txt
    mv biquandle.out quandle-related-birack-k-not-quandle.txt

11. Calculate the quandle-related biquandles that are not themselves quandles.

    biquandle-search -n=kXR quandle-related-birack-k-not-quandle.txt
    mv biquandle.out quandle-related-biquandle-k-not-quandle.txt

12. calculate virtual, weld and essential pairs from the biquandles and quandles. The file dominant-input-k.txt is a combination of the files biquandle-k-not-quandle.txt and quandle-k.txt

    dominant-pairs -n=k dominant-input-k.txt

The following steps were used to calculate the above lists for the cases k=5,6

1. Generate a list of distinct quandles of size k.

   nquandle -n=kT
   mv nquandle.out quandle-k.txt

2. Produce quandle related biracks from seed quandle patterns. Note: nswitch does not produce a distinct list.

   nswitch -n=kR quandle-k.txt
   mv nswitch.out quandle-related-birack-k-no-symmetry-check.txt

3. Reduce the quandle-related-biracks to a distinct list.

   biquandle-search -n=kbR quandle-related-birack-k-no-symmetry-check.txt
   mv biquandle.out quandle-related-birack-k.txt

4. Determine the set of quandle-related biquandles from the list of distinct biracks.

   biquandle-search -n=kRX quandle-related-birack-k.txt
   mv biquandle.out quandle-related-biquandle-k.txt

5. Remove the quandle-related biquandles from the list of distinct quandle-related biracks. The biquandle search that created quandle-related-biquandle-k.txt only added a condition to filter out biracks from quandle-related-birack-k.txt, so is a strict subset of that set of biracks, therefore we do not need to consider permuataions when removing them.

   biquandle-search -n=kbRXex quandle-related-birack-k.txt quandle-related-biquandle-k.txt
   mv biquandle.out quandle-related-birack-k-not-biquandle.txt

6. Determine the set of racks from the list of quandle-related biracks that are not biquandles. Since the identity appears in quandle-k.txt, quandle-related-birack-k.txt includes all the racks.

   biquandle-search -n=kXr quandle-related-birack-k.txt
   mv biquandle.out rack-k.txt

7. Remove the racks from the list of quandle-related biracks that are not biquandles.

   biquandle-search -n=kbRXe quandle-related-birack-k-not-biquandle.txt rack-k.txt
   mv biquandle.out quandle-related-birack-k-not-biquandle-not-rack.txt

8. Remove the quandles from the list of quandle-related-biquandles.

   biquandle-search -n=kRXe quandle-related-biquandle-k.txt quandle-k.txt
   mv biquandle.out quandle-related-biquandle-k-not-quandle.txt

9. Remove the quandles from the list of racks.

   biquandle-search -n=krXe rack-k.txt quandle-k.txt
   mv biquandle.out rack-k-not-quandle.txt

10. Remove the quandles from the quandle-related biracks (nswitch will have found the quandles from the seed U patterns).

    biquandle-search -n=kbRXe quandle-related-birack-k.txt quandle-k.txt
    mv biquandle.out quandle-related-birack-k-not-quandle.txt

11. calculate virtual, weld and essential pairs from the biquandles and quandles. The file dominant-input-k.txt is a combination of the files quandle-related-biquandle-k-not-quandle.txt and quandle-k.txt

    dominant-pairs -n=k dominant-input-k.txt

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