Good involutions of conjugation subquandles

TL;DR

This paper classifies and analyzes good involutions in conjugation subquandles, providing bounds, algorithms, and classifications that advance understanding in quandle theory and its applications to surface-knot theory.

Contribution

It offers a classification of good involutions in conjugation subquandles, develops algorithms for their computation, and classifies related structures up to order 8.

Findings

Sharp bounds on the number of good involutions in racks.

Algorithms implemented to compute all good involutions up to order 23.

Classification of symmetric and Legendrian racks and quandles up to order 8.

Abstract

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.