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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The eigenvalue equation does not imply the matrix relation

Statement refuted

Let $U, V \in S L_{2} (C)$ commute and have a common invariant line on which their eigenvalues are $M, L$ , respectively. For every primitive pair $(p, q) \in Z^{2}$ , if $M^{p} L^{q} = 1$ , then $U^{p} V^{q} = I_{2}$ .

Statement references

Exact matrix relation for a filling representation

Common peripheral eigenline and eigenvalue convention

Primitive filling-pair convention

Baldwin and Sivek, §2.3

Bénard, Florens, and Rodau, §2.2

Facts & Assumptions

Given: $N = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array})$ , $U = I_{2} + N$ , $V = I_{2}$ , and the primitive pair $(p, q) = (1, 0)$ .

[F1]

$N^{2} = 0$ , so $U$ is upper-triangular unipotent with determinant one and is not the identity.

[F2]

The identity matrix commutes with every matrix, and the eigenvalues of an upper-triangular matrix are its diagonal entries.

Counterexample

1.1F1F2givenalgebra

Since $N^{2} = 0$ , $U = (\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array})$ , so $U \in S L_{2} (C)$ , has only eigenvalue $1$ , and is not $I_{2}$ . Also $V = I_{2} \in S L_{2} (C)$ . By [F2], the pair commutes and has common eigenline $C (1, 0)$ with eigenvalues $M = L = 1$ .

2.1F1givenstep 1.1algebra∎

For $(p, q) = (1, 0)$ , the eigenvalue equation gives $M^{p} L^{q} = 1^{1} 1^{0} = 1$ , while $U^{p} V^{q} = U \neq I_{2}$ . This refutes the statement. The example is only a matrix pair and does not claim to extend to any knot-group representation.

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Sources

John A. Baldwin and Steven Sivek, Torus knots, the A-polynomial, and SL(2,C), §2.3 (standard reference, not scraped)
Léo Bénard, François Florens, and Adrien Rodau, A slope invariant and the A-polynomial of knots, §2.2 (standard reference, not scraped)