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

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SL2 character variety and peripheral eigenvalues

Definition

Assume full AC. Let $G = π_{1} (E_{K})$ be the group of a smooth oriented knot exterior, with positive meridian $μ$ and preferred longitude $λ$ . For a finite presentation $G = ⟨ g_{1}, \dots, g_{n} ∣ r_{1}, \dots, r_{m} ⟩$ , the $S L_{2} (C)$ representation variety is the algebraic set of tuples $(A_{1}, \dots, A_{n}) \in S L_{2} (C)^{n}$ for which every relator matrix $r_{j} (A_{1}, \dots, A_{n})$ equals the identity. The group $S L_{2} (C)$ acts on this set by simultaneous conjugation. Its affine categorical quotient, whose coordinate ring is the invariant ring of regular functions, is the $S L_{2} (C)$ character variety $X (E_{K})$ . For a representation $ρ$ , its character is the function $γ \mapsto tr (ρ (γ))$ . A representation is reducible if its matrices preserve a common one-dimensional subspace of $C^{2}$ ; otherwise it is irreducible. The peripheral torus group is $π_{1} (T_{K}) ≅ Z^{2}$ , generated by the meridian and preferred longitude; its image in $G$ is generated by their commuting classes and need not be isomorphic to $Z^{2}$ . Their image under any representation is a commuting pair of matrices, so it has a common invariant line and is simultaneously triangularizable over $C$ . Choosing a common invariant line gives meridian and longitude eigenvalues $(M, L) \in (C^{*})^{2}$ ; the unordered boundary character identifies $(M, L)$ with $(M^{- 1}, L^{- 1})$ . Simultaneous diagonalizability is stronger than simultaneous triangularizability.

Definition references

The finite knot-group presentation and positive meridians

The chosen compact exterior

The meridian and preferred longitude

Commuting peripheral matrices have a common invariant line

Cooper, Culler, Gillet, Long, and Shalen, §2

Tillmann, §§4.2–4.8

Depends on

Used by

The eigenvalue equation does not imply the matrix relation Counterexample
Adjoint-twisted homology and Culler-Shalen data Definition
Character-variety logarithmic slope Definition
The ordinary knot A-polynomial Definition
Figure-eight A-polynomial calculation Example
Baldwin–Sivek instanton-surgery and Newton-polygon input module Lemma
Figure-eight character curve and longitude trace Lemma
Finite trace-coordinate model on the stated loci Lemma
The peripheral eigenvalue locus is defined over Q Lemma
The SL2 representation variety is affine Lemma
Trace functions are regular and conjugation invariant Lemma
Trace-coordinate algebra for the SL2 character quotient Lemma
Boundary restriction and the surgery detector Theorem
The enhanced A-polynomial Newton polygon detects torus knots Theorem

Dependency tree · two levels

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Sources

D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen, Plane Curves Associated to Character Varieties of 3-Manifolds, §2 (standard reference, not scraped)
Stephan Tillmann, Varieties Associated to 3-Manifolds, §§4.2–4.8 (standard reference, not scraped)