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

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The trefoil group is nonabelian with abelianization Z

Statement

Let $T$ be the trefoil represented by the standard oriented three-crossing diagram below. Choose a basepoint $x_{0} \in S^{3} ∖ T$ in the diagram's upper collar and use the common upper-collar paths from $x_{0}$ to base the positive-meridian generators. Its under-arcs are labelled $a, b, c$ so that the ordered triples $(o, L, R)$ of over-arc, upper target under-arc and lower source under-arc are $(b, c, a)$ , $(c, a, b)$ , and $(a, b, c)$ at the three crossings; read each triple in the orientation-preserving local chart that straightens the over-arc to point rightward. The knot group $G_{T} = π_{1} (S^{3} ∖ T, x_{0})$ , where $π_{1}$ is based loops modulo endpoint-fixed homotopy with multiplication by concatenation, has presentation $⟨ a, b ∣ a b a = b a b ⟩$ . This presentation denotes the free group on $a, b$ modulo the normal closure of the displayed relation. Its abelianization, the quotient by its commutator subgroup, is isomorphic to $Z$ , but $G_{T}$ is nonabelian.

The drawing shows two oriented strands crossing three times; the two bottom endpoints join to the corresponding top endpoints by the outside arcs. The white gaps mark underpasses, and each redrawn strand is over. The crossing triples in the statement fix the generator labels and local chart convention.

Statement references

Wirtinger presentation theorem

Local crossing conjugacy convention

Cattabriga and Francaviglia, Knot Theory and Applications: Lecture Notes, §4.2

Charbonneau, Computing the Wirtinger Presentation of a Knot Group from a Knot Diagram, §§3–4.1

Facts & Assumptions

Given: The standard oriented trefoil diagram, crossing labels, chosen complement basepoint, and common upper-collar paths specified in the Statement.

[F1]

For a valid finite knot diagram with chosen complement basepoint $x_{0}$ , the Wirtinger presentation with one positive-meridian generator for each under-arc and one crossing conjugacy relator for each crossing presents $π_{1} (S^{3} ∖ T, x_{0})$ using the common upper-collar paths. Correctness of the Wirtinger presentation

[F2]

With a rightward over-strand and the common upper-collar basing paths, the upper target meridian $m_{L}$ and lower source meridian $m_{R}$ satisfy $m_{R} = m_{o} m_{L} m_{o}^{- 1}$ . Local meridian conjugacy at a Wirtinger crossing

[F3]

A group presentation $⟨ a, b ∣ r ⟩$ denotes the free group on $a, b$ modulo the normal closure of $r$ .

Proof

1.1F1F2F3givenalgebra

Applying [F2] to the three crossing triples gives $a = b c b^{- 1}$ , $b = c a c^{- 1}$ , and $c = a b a^{- 1}$ . Substitute the third equation into the first to obtain $a = b a b a^{- 1} b^{- 1}$ ; right multiplication by $b a$ gives $a b a = b a b$ . Under this relation, $c a c^{- 1} = a b a^{- 1} a a b^{- 1} a^{- 1} = a b a b^{- 1} a^{- 1} = b a b b^{- 1} a^{- 1} = b$ , so the second crossing equation follows. Therefore eliminating $c$ from the presentation in [F1] yields exactly $⟨ a, b ∣ a b a = b a b ⟩$ .

2.1F3step 1.1algebra

In the abelianization, the relation becomes $2 [a] + [b] = [a] + 2 [b]$ , hence $[a] = [b]$ . The abelianization is the free abelian group on $[a], [b]$ modulo this single relation, so it is $Z$ .

3.1step 1.1algebra∎

Map $a \mapsto (12)$ and $b \mapsto (23)$ in $S_{3}$ . Both $a b a$ and $b a b$ map to $(13)$ , so the relation is satisfied. These two transpositions generate $S_{3}$ , making the homomorphism surjective. Since $S_{3}$ is nonabelian, its surjective preimage $G_{T}$ is nonabelian.

Depends on

Used by

Abelianized peripheral data do not distinguish the unknot from the trefoil Counterexample

Dependency tree · two levels

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Sources

Alessia Cattabriga and Stefano Francaviglia, Knot Theory and Applications: Lecture Notes, §4.2 (standard reference, not scraped)
Cole Charbonneau, Computing the Wirtinger Presentation of a Knot Group from a Knot Diagram, §§3–4.1 (standard reference, not scraped)