Good involutions of conjugation subquandles

Posed by Taniguchi, the classification of quandles with good involutions is a difficult question with applications to surface-knot theory. We address this question for subquandles of conjugation quandles, including all core quandles. We also study good involutions of faithful racks. In particular, we obtain sharp bounds on the number of good involutions of racks in these families. As an application of our results, we implement group-theoretic algorithms that compute all good involutions of conjugation quandles and core quandles; we provide data for those up to order 23. As another application, we construct infinite families of connected, noninvolutory symmetric quandles. We also classify symmetric and Legendrian racks, quandles, and kei up to order 8 using a computer search. Finally, we exhibit an equivalence of categories between racks and Legendrian racks that induces an equivalence between involutory racks, Legendrian kei, and symmetric kei.