LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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The dihedral operation is a quandle

Statement

For an odd prime $p$ , let $F_{p} = Z / p Z$ and define $a * b = 2 b - a$ . Then $(F_{p}, *)$ is a quandle.

Statement references

Quandles and knot-diagram colorings.

Manturov, Knot Theory, Chapter 5.

Carter, A Survey of Quandle Ideas.

Facts & Assumptions

Given: An odd prime $p$ and $a, b, c \in F_{p}$ , with $a * b = 2 b - a$ .

[F1]

Quandles and knot-diagram colorings: a quandle has bijective right translations, is idempotent, and is right self-distributive.

[A1]

$p$ is prime, so a nonzero residue class modulo $p$ has an inverse.

Proof

1.1givenA1construct

Addition and multiplication of residue classes are well-defined because congruence modulo $p$ is preserved by sums and products. If $\overset{ˉ}{r} \neq 0$ in $Z / p Z$ , choose an integer representative $r$ not divisible by $p$ . Since $p$ is prime, $\gcd (r, p) = 1$ ; Bézout's identity gives integers $u, v$ with $u r + v p = 1$ , so $\overset{ˉ}{u} \overset{ˉ}{r} = 1$ . Hence the residue classes form a field. For fixed $b$ , the right translation is $R_{b} (a) = 2 b - a$ and satisfies $R_{b} (R_{b} (a)) = 2 b - (2 b - a) = a$ . It is therefore a bijection.

1.2givenF1algebra

Idempotency holds because $a * a = 2 a - a = a$ .

2.1F1step 1.1step 1.2algebra∎

For all $a, b, c$ , $(a * b) * c = 2 c - (2 b - a) = a - 2 b + 2 c$ , while $(a * c) * (b * c) = 2 (2 c - b) - (2 c - a) = a - 2 b + 2 c$ . Thus right self-distributivity holds; together with step 1.1 and step 1.2, [F1] gives the quandle axioms.

Sources

Quandles and knot-diagram colorings.

Depends on

Quandles and knot-diagram colorings

Used by

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

V. O. Manturov, Knot Theory, Chapter 5, §§5.1–5.2, printed pp. 49–55 (standard reference, not scraped)
J. Scott Carter, A Survey of Quandle Ideas, §2, arXiv:1002.4429v2, pp. 3–5 (standard reference, not scraped)