Cyclic-cover homology order from the Alexander polynomial

Given: Full AC, a smooth oriented knot $K$, and $q\ge 2$.

Full AC is the choice-function axiom required by the root-product lemma. The Axiom of Choice

The root-product lemma gives the finiteness criterion and order for any primitive $q$-th root. Root-product test for cyclic branched-cover homology

${\mathrm{\Delta }}_K(t)$ is defined up to multiplication by $\pm t^k$. The Alexander polynomial from the zeroth elementary ideal

Full AC supplies the countable-choice hypotheses used by the surface-existence and surface-normal-form results: apply its choice function to the range of any sequence of nonempty sets and reuse the selected element at each index. Thus the sequence assertions in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) and The countable-choice principle used in the foliation pair hold. The Axiom of Choice

Every smooth oriented knot bounds a connected compact smooth oriented embedded surface with its given boundary, which is a Seifert surface under Oriented Seifert surfaces in the three-sphere. Every smooth oriented knot in $S^3$ bounds a Seifert surface For a connected compact oriented surface with one boundary component and genus $g$, the fundamental group is free of rank $2g$. Finite surface normal forms, Jordan disks, and torsion control

A connected smooth surface $F$ is locally path-connected, so its path components are open and it is path-connected. The first Hurewicz map identifies $H_1(F;\mathbb{Z})$ with the abelianization of ${\pi }_1(F)$; for the genus-$g$ surface in [F4], this gives $H_1(F;\mathbb{Z})\cong {\mathbb{Z}}^{2g}$ and an ordered integral basis. The first Hurewicz map is abelianization