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  "body": "\n## Example\n\nLet $n \\ge 1$ and let $R_n = \\Z/n\\Z$ be the [[integers modulo $n$|03VFZ2WETT]], with\n$$a \\triangleright b = 2a - b.$$\nThen $(R_n, \\triangleright)$ is a [[quandle|X74VY86TSX]], the *dihedral quandle* of order $n$.\n\nCheck of the three axioms:\n\n1. Self-distributivity: $a \\triangleright (b \\triangleright c) = 2a - 2b + c$ and $(a \\triangleright b) \\triangleright (a \\triangleright c) = 2(2a - b) - (2a - c) = 2a - 2b + c$.\n2. For fixed $a$ and $b$, the equation $a \\triangleright c = b$ has the unique solution $c = 2a - b$.\n3. Idempotence: $a \\triangleright a = 2a - a = a$.\n\nThe name comes from the [[dihedral group|0JZ27DMJ5D]] $D_n$.\nLet $s_a$ be the reflection $x \\mapsto 2a - x$ of $\\Z/n\\Z$.\nThen $s_a s_b s_a^{-1} = s_{2a - b}$, so the operation of $R_n$ is conjugation of reflections.\nFor odd $n$ the map $a \\mapsto s_a$ is injective, and $R_n$ is the conjugacy class of reflections in $D_n$ with conjugation.\n\n## Remarks\n\nColourings of a [[knot|QYEAZTYGPT]] diagram by $R_3$ are the Fox $3$-colourings: at each crossing, the three arcs carry either one colour or three different colours.\nThe trefoil has $9$ colourings by $R_3$, three of them constant, while the unknot has only the $3$ constant ones.\nSince the number of colourings is a knot invariant, the trefoil is knotted.\nIn Mathlib this quandle is `Quandle.Dihedral`.\n\n## Explanations\n\n### L1 human\n\nPlace $n$ points evenly on a circle and reflect the circle in the line through point $a$ and the centre.\nPoint $b$ moves to point $2a - b$.\nThe dihedral quandle records exactly this rule.\nThese simple quandles already detect interesting knots, through the familiar game of colouring a knot diagram with three colours.\n",
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