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

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The enhanced A-polynomial

Definition

Assume full AC and let $K \subset S^{3}$ be a smooth oriented knot. The enhanced A-polynomial ${\tilde{A}}_{K} (M, L)$ is the primitive square-free Laurent polynomial defining the union of one-dimensional peripheral-eigenvalue image components arising from irreducible characters of $π_{1} (E_{K})$ . It is defined up to multiplication by $\pm M^{a} L^{b}$ . The ordinary A-polynomial is the least common multiple $A_{K} = lcm ({\tilde{A}}_{K}, L - 1)$ , up to a Laurent unit, where the least common multiple uses each distinct irreducible Laurent factor only once; the factor $L - 1$ is the contribution of the abelian character component, and an irreducible component may also contribute that same factor.

Definition references

The ordinary A-polynomial and its abelian component

Baldwin and Sivek, §2.3

Ni and Zhang, §2

Depends on

The ordinary knot A-polynomial

Used by

Enhanced A-polynomial and Alexander degree identify torus knots Corollary
Character-variety logarithmic slope Definition
Figure-eight A-polynomial calculation Example
Baldwin–Sivek instanton-surgery and Newton-polygon input module Lemma
Enhanced A-polynomial of a torus knot Lemma
A-polynomial Newton parallelogram characterization Open problem
A-polynomial questions for knots in other 3-manifolds Open problem
Jones polynomial unknot-detection status Remark
Boundary restriction and the surgery detector Theorem
Enhanced A-polynomial Newton edges give strict boundary slopes Theorem
The enhanced A-polynomial Newton polygon detects torus knots Theorem

Dependency tree · two levels

7 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)
Yi Ni and Xingru Zhang, Characterizing slopes for torus knots, §2 (standard reference, not scraped)