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

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The SL2 representation variety is affine

Statement

Assume full AC and let $G = π_{1} (E_{K}) = ⟨ g_{1}, \dots, g_{n} ∣ r_{1}, \dots, r_{m} ⟩$ be a finite presentation of a knot-exterior group. The set of homomorphisms $G \to S L_{2} (C)$ is the affine algebraic subset of $S L_{2} (C)^{n}$ cut out by the matrix-entry equations $r_{j} (A_{1}, \dots, A_{n}) = I_{2}$ for $1 \leq j \leq m$ . After each inverse letter is replaced by the adjugate of its determinant-one matrix, these equations have integer coefficients in the matrix-entry coordinates.

Statement references

Bénard, Florens, and Rodau, §2.1

Tillmann, §4.2

Facts & Assumptions

Given: Full AC and the finite presentation $G = ⟨ g_{1}, \dots, g_{n} ∣ r_{1}, \dots, r_{m} ⟩$ .

[F1]

The knot group has a finite Wirtinger presentation (Correctness of the Wirtinger presentation).

[F2]

If $A = (\begin{array}{cc} a & b \\ c & d \end{array}) \in S L_{2} (C)$ , then $A^{- 1} = (\begin{array}{cc} d & - b \\ - c & a \end{array})$ , an integer-polynomial expression in its entries: multiplying by the displayed matrix gives $\det (A) I = I$ entrywise.

Proof

1.1F1givenalgebra

Work with the fixed finite presentation in the Statement. Fact [F1] ensures the existence of finite presentations for the knot group, but no change from the specified presentation to different Wirtinger generators is presumed here. Write each generator matrix as four coordinate variables and impose the polynomial determinant equation $\det (A_{i}) = 1$ . These finitely many equations cut out $S L_{2} (C)^{n}$ in affine $4 n$ -space.

2.1F2step 1.1algebra

Evaluate each relator word by matrix multiplication, replacing each inverse letter by the adjugate matrix from [F2]. Every entry of $r_{j} (A_{1}, \dots, A_{n}) - I_{2}$ is therefore a polynomial in the coordinates.

3.1givenstep 1.1step 2.1algebra∎

A tuple in $S L_{2} (C)^{n}$ satisfies all these matrix equations exactly when its generator assignment extends to a homomorphism from the presented group $G$ . Hence their common zero set is precisely $Hom (G, S L_{2} (C))$ , a closed algebraic subset of affine space.

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Sources

Léo Bénard, François Florens, and Adrien Rodau, A slope invariant and the A-polynomial of knots, §2.1 (standard reference, not scraped)
Stephan Tillmann, Varieties associated to 3-manifolds, §4.2 (standard reference, not scraped)