Quandles of cyclic type with several fixed points

TL;DR

This paper classifies and characterizes cyclic type quandles with multiple fixed points, proving uniqueness results for connected cases and providing conditions for their existence and construction.

Contribution

It establishes the uniqueness of connected quandles of cyclic type with fixed points and characterizes the range and construction of non-connected quandles with similar properties.

Findings

Unique connected quandle of order 6 with 2 fixed points (octahedron quandle)

Non-connected quandles occur only within specific order ranges

Conditions and methods for constructing new quandles from existing ones

Abstract

A quandle of cyclic type of order $n$ with $f\geq 2$ fixed points is such that each of its permutations splits into $f$ cycles of length $1$ and one cycle of length $n-f$. In this article we prove that there is only one such connected quandle, up to isomorphism. This is a quandle of order $6$ and $2$ fixed points, known in the literature as octahedron quandle. We prove also that, for each $f\geq 2$, the non-connected versions of these quandles only occur for orders $n$ in the range $f+2 \leq n \leq 2f$ and that, for each $f>1$, there is only one such quandle of order $2f$ with $f$ fixed points, up to isomorphism. Still in the range $f+2 \leq n \leq 2f$, we present sufficient conditions for the existence of such quandles, writing down their permutations; we also show how to obtain new quandles form old ones, leaning on the notion of common fixed point.