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.

The up and down actions contained in the lists on this page correspond to the maps s^a and s_a. The lists are defined up to isomorphism: permutation symmetries (conjugation), inverse (interchange of classical crossing sign), interchange of the up and down action (orientation reversal), and combinations thereof, as described in reference [1] on the finite biquandles page.

• 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 essential virtual pairs of size 2
• There are no essential welded pairs of size 2
• 2 virtual invariant pairs of size 2 that are not essential virtual pairs or essential welded pairs


• 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
• 3 essential virtual pairs of size 3
• 2 essential welded pairs of size 3
• 18 virtual invariant pairs of size 3 that are not essential virtual pairs or essential welded pairs


• 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
• 37 essential virtual pairs of size 4
• 8 essential welded pairs of size 4
• 298 virtual invariant pairs of size 4 that are not essential virtual pairs or essential welded pairs


• 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
• 34 quandle-related essential virtual pairs of size 5
• 7 quandle-related essential welded pairs of size 5
• 471 quandle-related virtual invariant pairs of size 5 that are not essential virtual pairs or essential welded pairs


• 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
• 527 quandle-related essential virtual pairs of size 6
• 40 quandle-related essential welded pairs of size 6
• 9846 quandle-related virtual invariant pairs of size 6 that are not essential virtual pairs or essential welded pairs

The essential welded 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=kbS
   mv biquandle.out birack-k.txt

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

   biquandle-search -n=kRXS 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=kbRXexS 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=krS 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=kbRXeS 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=kqS birack-k.txt
   mv biquandle.out quandle-k.txt

7. Remove the quandles from the list of biquandles.

   biquandle-search -n=kRXeS 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=krXeS 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=kbRQDS birack-k.txt
   mv biquandle.out quandle-related-birack-k.txt

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

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

12. calculate virtual, weld and essential welded 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=kbRS 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=kRXS 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=kbRXexS 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=kXrS 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=kbRXeS 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=kRXeS 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=krXeS 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=kbRXeS quandle-related-birack-k.txt quandle-k.txt
    mv biquandle.out quandle-related-birack-k-not-quandle.txt

11. calculate virtual, weld and essential welded 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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