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

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Trace functions are regular and conjugation invariant

Statement

Assume full AC. For every word $w$ in the generators of a finitely presented group $G$ , the function $ρ \mapsto tr (ρ (w))$ is a regular function on $Hom (G, S L_{2} (C))$ and is invariant under simultaneous conjugation.

Statement references

Bénard, Florens, and Rodau, §2.1

Tillmann, §§4.2–4.4

Facts & Assumptions

Given: Full AC, a finite presentation of $G$ , its affine representation variety, and a word $w$ in the generators and their inverses.

[F1]

For the given finite presentation, impose determinant-one equations and the matrix-entry relator equations in the finite generator matrix space. By the inverse formula [F2], these are polynomial equations for every finitely presented group, not just a knot group.

[F2]

If $A = (\begin{array}{cc} a & b \\ c & d \end{array}) \in S L_{2} (C)$ , then $A^{- 1} = (\begin{array}{cc} d & - b \\ - c & a \end{array})$ , as verified by direct multiplication using $a d - b c = 1$ .

Proof

1.1F1F2constructalgebra

The equations of [F1] cut out exactly the representation tuples, since a generator assignment satisfies them precisely when it respects every presented relator. Evaluate $w$ by multiplying its generator matrices and the adjugates for inverse letters from [F2]. Each entry of the resulting matrix is a polynomial in the representation coordinates, so its trace is regular.

2.1step 1.1algebra

For any $P \in S L_{2} (C)$ , conjugation sends the word matrix $W$ to $P W P^{- 1}$ . The identity $tr (P W P^{- 1}) = tr (W P^{- 1} P) = tr (W)$ proves invariance.

3.1step 1.1step 2.1∎

Thus each word trace defines an element of the conjugation-invariant coordinate ring of the representation variety.

Depends on

Used by

Trace-coordinate algebra for the SL2 character quotient Lemma

Dependency tree · two levels

8 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

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, §§4.2–4.4 (standard reference, not scraped)