More Automorphism Groups of Quandles
Quinn J. M. Salix, Peyton Phinehas Wood
TL;DR
This work advances the study of finite quandle automorphisms by giving a complete description of the displacement group for dihedral quandles: $Dis(R_n) \cong \mathbb{Z}_n$ when $n$ is odd and $Dis(R_n) \cong \mathbb{Z}_{n/2}$ when $n$ is even. It also proves $Dis(X) \cong Inn(X)$ for quandles with at least one trivial column, and for one non-trivial column quandles $Dis(P_n^{\sigma}) = Inn(P_n^{\sigma}) \cong \mathbb{Z}_{|\sigma|}$. The paper further confirms and extends known automorphism and inner automorphism groups for small quandles, and provides comprehensive computations up to order $|X| \le 10$ using GAP- and Sage-based workflows, building on the Vor\'Yang enumeration up to order $13$. These results enhance our understanding of quandle symmetries and furnish extensive data for knot-theoretic applications where quandle automorphisms act as invariants and structural constraints.
Abstract
We prove that the displacement group of the dihedral quandle with n elements is isomorphic to the group generated by rotations of the n/2-gon when n is even and the n-gon when n is odd. We additionally show that any quandle with at least one trivial column has equivalent displacement and inner automorphism groups. Then, using a known enumeration of quandles which we confirm up to order 10, we verify the automorphism group and the inner automorphism group of all quandles (up to isomorphism) of orders less than or equal to 7, compute these for all 115,431 quandles orders 8, 9, and 10, and extend these results by computing the displacement group of all 115,837 quandles (up to isomorphism) of order less than or equal to 10.