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Enhanced A-polynomial and Alexander degree identify torus knots

Statement

Assume full AC. Let $a \geq b \geq 1$ be coprime positive integers, and let $T_{a, b} \subset S^{3}$ be the corresponding torus knot. If a knot $K \subset S^{3}$ has the same enhanced A-polynomial as $T_{a, b}$ up to a Laurent unit and the same Alexander-polynomial breadth (highest exponent minus lowest exponent of its symmetrized Alexander polynomial), then $K$ has the same underlying unoriented knot type as $T_{a, b}$ . When $b \geq 2$ , the absolute torus parameters of $K$ agree with ${a, b}$ .

Statement references

Thin enhanced-A classification

Alexander breadth formula for torus knots

Baldwin and Sivek, Corollary 1.4 and §4

Facts & Assumptions

Given: Full AC, a knot $K$ , a positive torus knot $T_{a, b}$ , equality of enhanced A-polynomials up to a Laurent unit, and equality of Alexander breadths.

[F1]

A knot whose enhanced A-polynomial has Newton polygon contained in a line is a torus knot. For a positive nontrivial torus knot $T_{a, b}$ with $a > b \geq 2$ , its enhanced A-polynomial has a line-segment Newton polygon of slope $a b$ in the stated convention; the unknot has enhanced A-polynomial $1$ (The enhanced A-polynomial Newton polygon detects torus knots; Enhanced A-polynomial of a torus knot). The same segment-slope formula holds for signed parameters: the mirror changes peripheral coordinates by $(M, L) \mapsto (M^{- 1}, L)$ , reversing the knot orientation changes them by $(M, L) \mapsto (M^{- 1}, L^{- 1})$ , and these transformations are proved for the enhanced locus in The enhanced A-polynomial Newton polygon detects torus knots, Proof2.1. They send the positive segment slope $∣ p q ∣$ to the signed product $p q$ .

[F2]

For signed coprime parameters, the Alexander breadth of $T_{p, q}$ is $(∣ p ∣ - 1) (∣ q ∣ - 1)$ ; changing both signs reverses knot orientation and changing one sign takes the mirror (Torus-knot Alexander polynomial and genus formulas; Coprime torus knot).

[F3]

A torus knot with one parameter of absolute value $1$ is the standard unknot (Unit-slope torus knots are unknots).

[A1]

Full AC implies countable choice, the hypothesis used by [F3] (The Axiom of Choice; AC implies DC implies countable choice).

[F4]

The classification used in [F1] is now complete with the explicit external input Baldwin–Sivek, arXiv:2405.19197v2, Theorem 1.2. Their Corollary 1.4 (p.2) independently states detection by the enhanced polynomial together with Alexander degree; the finite parameter-recovery proof below is retained locally. The source statement was consulted directly, not its gauge-theoretic proof reconstructed. The breadth used here is twice the degree of the symmetric polynomial for torus knots, so equality of breadths gives exactly the equality used in step2.1.

Proof

1.1F1givencasesalgebra

Equality of enhanced A-polynomials up to a Laurent unit gives the same Newton polygon shape. By [F1], this polygon lies in a line, so write $K = T_{p, q}$ for signed coprime nonzero parameters. If $b \geq 2$ , both polygons are line segments and [F1] gives equality of slopes, hence $p q = a b$ . If $b = 1$ , the target polygon is a point and no slope equation is asserted.

2.1F2step 1.1algebra

Suppose $b \geq 2$ and put $u = ∣ p ∣$ , $v = ∣ q ∣$ . From $p q = a b > 0$ , $p$ and $q$ have the same sign and $u v = a b$ . Equality of Alexander breadths and [F2] give $(u - 1) (v - 1) = (a - 1) (b - 1)$ . Expanding and substituting $u v = a b$ yields $u + v = a + b$ .

2.2A1F2F3step 1.1givenalgebra

Suppose $b = 1$ . Then [F3] says $T_{a, 1}$ is the unknot, whose Alexander breadth is zero. By [F2] and the assumed breadth equality, $(∣ p ∣ - 1) (∣ q ∣ - 1) = 0$ . Thus $∣ p ∣ = 1$ or $∣ q ∣ = 1$ , and [A1] supplies the countable-choice hypothesis of [F3], so $K = T_{p, q}$ is the unknot as well. Hence its underlying unoriented knot type equals $T_{a, 1}$ .

3.1step 2.1algebra

The monic quadratics $x^{2} - (u + v) x + u v$ and $x^{2} - (a + b) x + a b$ are equal, so their root multisets agree and ${u, v} = {a, b}$ . The same-sign conclusion in step 2.1 means that $K$ is $T_{a, b}$ up to orientation, so the underlying unoriented knot types agree.

4.1step 3.1step 2.2∎

Steps 3.1 and 2.2 cover respectively the nontrivial and unknot target cases, proving the stated identification; in the nontrivial case step 3.1 also identifies the absolute parameters up to interchange.

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Sources

John A. Baldwin and Steven Sivek, Torus knots, the A-polynomial, and SL(2,C), Corollary 1.4 and §4 (standard reference, not scraped)