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

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The ordinary knot A-polynomial

Definition

Assume full AC and let $K \subset S^{3}$ be a smooth oriented knot with exterior $E_{K}$ , positive meridian $μ$ , and preferred longitude $λ$ . Define $E_{K} \subset (C^{*})^{2}$ to consist of all pairs $(M, L)$ for which some representation $ρ : π_{1} (E_{K}) \to S L_{2} (C)$ has a common invariant line on which $ρ (μ)$ acts by $M$ and $ρ (λ)$ acts by $L$ . The Zariski closure is the smallest Zariski-closed algebraic subset containing $E_{K}$ ; let $V_{K}$ be the union of its one-dimensional irreducible components, where an irreducible component is a maximal irreducible closed subset. Zero-dimensional image components are omitted. The ordinary A-polynomial $A_{K} (M, L)$ is the Laurent polynomial generating the vanishing ideal of $V_{K}$ , chosen square-free (no repeated irreducible factors) and with primitive integer coefficients (their greatest common divisor is one). It is determined up to multiplication by $\pm M^{a} L^{b}$ for $a, b \in Z$ . Its reduced curve includes the abelian component $L = 1$ ; equivalently, its representative in $Z [M, L]$ is divisible by $L - 1$ .

Definition references

Character variety and peripheral eigenvalues

Exterior homology and the preferred longitude

Baldwin and Sivek, §2.3

Cooper, Culler, Gillet, Long, and Shalen, §§2–3

Depends on

Used by

Character-variety logarithmic slope Definition
The enhanced A-polynomial Definition
Figure-eight A-polynomial calculation Example
Baldwin–Sivek instanton-surgery and Newton-polygon input module Lemma
Boundary eigenvalue curve elimination for the A-polynomial Lemma
Rationality of the adjoint-twisted slope Lemma
A-polynomial Newton parallelogram characterization Open problem
A-polynomial questions for knots in other 3-manifolds Open problem
Does the ordinary A-polynomial detect torus knots? Open problem
Jones polynomial unknot-detection status Remark
Adjoint logarithmic Gauss slope on an affine character component Theorem
Bénard–Florens–Rodau logarithmic Gauss formula and ideal-point slope Theorem
Boundary restriction and the surgery detector Theorem
Newton polygon edge slopes are boundary slopes Theorem
The enhanced A-polynomial Newton polygon detects torus knots Theorem

Dependency tree · two levels

27 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

John A. Baldwin and Steven Sivek, Torus knots, the A-polynomial, and SL(2,C), §2.3 (standard reference, not scraped)
Cooper, Culler, Gillet, Long, and Shalen, Plane curves associated to character varieties of 3-manifolds, §§2–3 (standard reference, not scraped)