DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Racks, diagram fundamental quandles, and adjoint groups

Definition

A rack is a set $R$ with a binary operation $* : R \times R \to R$ such that for every $y \in R$ the right translation $R_{y} (x) = x * y$ is a bijection and $(x * y) * z = (x * z) * (y * z)$ for all $x, y, z \in R$ . A quandle is a rack satisfying $x * x = x$ for every $x$ . For either structure, write $x \overset{ˉ}{*} y = R_{y}^{- 1} (x)$ .

For a regular oriented link diagram $D$ , let $A (D)$ be its finite set of maximal under-arcs. For a rack $R$ , an $R$ -coloring of $D$ is an assignment $c : A (D) \to R$ such that at every crossing with source under-arc $u$ , target under-arc $v$ , and overarc $o$ , $c (v) = c (u) * c (o)$ , using the positive-normal convention of the diagram-coloring definition. Write ${Col}_{R} (D)$ for the set of these assignments. When $R$ is a quandle, this agrees with the quandle-coloring definition.

Form the set of finite terms recursively: each symbol $x_{a}$ for $a \in A (D)$ is a term, and if $s, t$ are terms then $s * t$ and $s \overset{ˉ}{*} t$ are terms. Let $\sim$ be the intersection of all equivalence relations on these terms that are compatible with both operations and contain the quandle identities: idempotency, right invertibility, and right self-distributivity, together with the crossing relations. At a crossing with source and target under-arcs $u, v$ and overarc $o$ , the crossing relation is $x_{v} = x_{u} * x_{o}$ ; reversing source and target gives the equivalent relation with $\overset{ˉ}{*}$ . The quotient term set $Q (D)$ is the diagram quandle, or the diagram presentation of the fundamental quandle. Its independence of diagram is proved in the Theorem.

For any quandle $Q$ , its adjoint group $Adj (Q)$ is the group presentation with generators $e_{q}$ for all $q \in Q$ and relators $e_{q * r} = e_{r}^{- 1} e_{q} e_{r}$ for all $q, r \in Q$ . A group presentation $⟨ X ∣ S ⟩$ means the quotient of the free group on $X$ by the normal closure of the relator words $S$ .

The diagram presentation and the adjoint group are algebraic constructions; this definition makes no completeness or knot-classification assertion.

Definition references

Quandles and knot-diagram colorings.

Regular oriented link diagrams.

Group presentation by generators and relations.

Diagram fundamental quandles and adjoint groups from Wirtinger data.

Manturov, Knot Theory, Chapter 5.

Carter, A Survey of Quandle Ideas.

Nelson, Link invariants from finite racks.

Depends on

Used by

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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, §§2.1 and 2.2, arXiv:1002.4429v2, pp. 3–8 (standard reference, not scraped)
Sam Nelson, Link invariants from finite racks, §§2.1–2.4, arXiv:0808.0029v3, pp. 2–6 (standard reference, not scraped)