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Enhanced A-polynomial of a torus knot

Statement

Assume full AC and let $a > b \geq 2$ be coprime positive integers. With the meridian-longitude and exponent conventions used for the enhanced A-polynomial, the enhanced A-polynomial of $T_{a, b}$ is, up to multiplication by a Laurent unit, ${\tilde{A}}_{T_{a, b}} (M, L) = {\begin{cases} 1 + M^{2 a} L, & b = 2, \\ - 1 + M^{2 a b} L^{2}, & b > 2. \end{cases}$ If $K$ is the unknot, then ${\tilde{A}}_{K} = 1$ . The nontrivial torus-knot Newton polygon is therefore a line segment whose slope in the $(a, b) \mapsto M^{b} L^{a}$ convention is $a b$ .

Statement references

Enhanced A-polynomial convention

Torus-knot group and peripheral words

Baldwin and Sivek, Example 2.13

Ni and Zhang, Lemma 2.1

Facts & Assumptions

Given: Full AC, the oriented positive coprime torus knot and original meridian-longitude and Newton-coordinate conventions.

[F1]

The actual torus-knot group/peripheral supplier gives $G = ⟨ x, y ∣ x^{a} = y^{b} ⟩$ , central $h = x^{a} = y^{b}$ , positive meridian $μ = y^{α} x^{β}$ with $a α + b β = 1$ , and preferred longitude $λ = h μ^{- a b}$ (Torus-knot exterior group and peripheral system).

[F2]

For exactly these group words, the complete irreducible peripheral eigenvalue-locus ideal is $(1 + M^{2 a} L)$ for $b = 2$ and $(M^{2 a b} L^{2} - 1)$ for $b > 2$ . The proof supplies explicit irreducible matrix families with a cofinite set of meridian values and proves the exact radical ideal, including exceptional representations (Irreducible peripheral eigenvalue curves of a torus presentation).

[F3]

The crossing-free unknot has group $Z$ , by the actual one-generator, no-relator calculation in Explicit unknot and figure-eight oriented Wirtinger codes.

[F4]

The enhanced A-polynomial defines the reduced union of one-dimensional peripheral eigenvalue-image closure components from irreducible characters, with primitive integer normalization (The enhanced A-polynomial).

Proof

1.1F1F2F4givenconstructalgebra

Apply [F1] with the given positive integers $a, b$ . Its meridian and longitude are exactly the words required in [F2], including the sign and order of the positive meridian. For $b = 2$ , [F2] gives a single graph $L = - M^{- 2 a}$ with exact ideal $(1 + M^{2 a} L)$ . For $b > 2$ it gives the two graphs $L = \pm M^{- a b}$ with exact ideal $(M^{2 a b} L^{2} - 1)$ . Each graph is a one-dimensional closed copy of $C^{*}$ under its meridian coordinate; the exact graph ideals in [F2] prove that no additional isolated or other-dimensional image component contributes. The reduced union in [F4] therefore has precisely the displayed equation. These integer polynomials are primitive because their nonzero coefficients are $1, - 1$ , and square-free because [F2] identifies the graph ideals as distinct prime ideals with radical intersection. Thus they have the asserted normalization up to a Laurent unit.

1.2F3F4givenalgebra

For the unknot, [F3] identifies a representation with a single determinant-one matrix. Its complex eigenvector supplies a line invariant under that matrix and all its positive and negative powers. Hence there is no irreducible representation of the unknot group. The defining enhanced curve union in [F4] is empty, with unit vanishing ideal, so its primitive generator is $1$ . This argument concerns only the unknot and makes no converse detection claim requiring instanton theory.

2.1F1step 1.1givenalgebra∎

In the specified exponent coordinates a monomial $M^{j} L^{i}$ corresponds to $(i, j)$ . The two terms for $b = 2$ give points $(0, 0)$ and $(1, 2 a)$ , so the segment has slope $2 a = a b$ . For $b > 2$ the points are $(0, 0)$ and $(2, 2 a b)$ , so the slope is again $a b$ . Multiplication by a Laurent unit translates both points and leaves the slope unchanged. The group/peripheral supplier retains its original full-AC hypothesis, whereas the new finite matrix calculation adds no choice principle. This proves every clause with the original positive-knot and peripheral conventions.

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Sources

John A. Baldwin and Steven Sivek, Torus knots, the A-polynomial, and SL(2,C), Example 2.13 (standard reference, not scraped)
Yi Ni and Xingru Zhang, Characterizing slopes for torus knots, Lemma 2.1 (standard reference, not scraped)