GAP program for classifying racks, quandles, and kei with good involutions up to order 11, with data up to order 8. Also contains programs for computing good involutions of conjugation quandles and core quandles, with data up to order 23.
Compiled by Lực Ta
Based on the rack library of Petr Vojtěchovský and Seung Yeop Yang (https://www.cs.du.edu/~petr/libraries_of_algebraic_structures.html)
See the paper "Good involutions of conjugation subquandles" for further information.
symmetric-rack-finder.txt exhaustively searches for virtual structures on all racks of a given order n in Vojtěchovský and Yang's library, searches for isomorphisms between them, and outputs a list of all virtual racks of order n up to isomorphism.
For example, to run the search for all n between 1 and 3, load Vojtěchovský and Yang's library and then run the following. The following assumes that you've saved symmetric-rack-finder.txt in the same folder as the rack library; if not, then replace "LRQ.path" below and in symmetric-rack-finder.txt with the path to wherever you've saved symmetric-rack-finder.txt.
racks:=[1, 2, 6, 19, 74, 353, 2080, 16023, 159526, 2093244, 36265070];
for n in [1..3] do ReadAsFunction(Concatenation(LRQ.path, "symmetric-rack-finder.txt"))()(n,racks[n]); od;
Here, racks is a list whose n-th entry is the number of isomorphism classes of racks of order n. The entries of racks are taken from OEIS sequence A181770.
conj-finder.txt, conj-subquandle-finder.txt, and core-finder.txt respectively compute good involutions of all nontrivial conjugation quandles Conj G, all subquandles Conj X of nontrivial conjugation quandles Conj G, and nontrivial core quandles Core G for all groups of a given order n. After reading these files into GAP, the functions are called by symmConj(n), symmConjSubs(n), and symmCore(n), respectively.
For example, to compute good involutions of conjugation quandles Conj G for all nonabelian groups G up to order 23, run the following. As discussed above, replace "LRQ.path" as needed.
Read(Concatenation(LRQ.path, "symmetric-rack-finder.txt"))()(n,racks[n]);
for n in [6..23] do symmConj(n); od;