Prime-power cyclic branched covers of knots have finite first homology

Given: Full AC, a smooth oriented knot $K$, and $q=p^k$ with $p$ prime and $k\ge 1$.

Full AC is assumed wherever the cover and orientation conventions use the choice axiom. The Axiom of Choice

For any connected Seifert surface with positive-push matrix $V$, the group $H_1({\mathrm{\Sigma }}_q(K);\mathbb{Z})$ is finite exactly when $D({\zeta }_q^j)\mathrm{\ne }0$ for all $1\le j<q$, where $D(t)=\det (V-t{V}^T)$. Root-product test for cyclic branched-cover homology

$D(1)=\det (V-V^T)=\pm 1$, since $V-V^T$ is the integral intersection matrix of the Seifert surface. The skew part of a Seifert matrix is the surface intersection form

For each $s\ge 1$, the minimal polynomial over $\mathbb{Q}$ of a primitive $p^s$-th root is the monic cyclotomic polynomial ${\mathrm{\Phi }}_{p^s}(t)$, which has integer coefficients and ${\mathrm{\Phi }}_{p^s}(1)=p$. ${\mathrm{\Phi }}_n$ is irreducible in $\mathbb{Q}[t]$ for every $n\ge 1$

If $Y$ is closed, connected, oriented and has finite first homology, its bounding-chain torsion linking pairing is nonsingular. The torsion linking pairing of a rational homology sphere is nonsingular

For a closed oriented $3$-manifold with finite first homology, the homology universal coefficient theorem and Poincaré duality give rational homology of $S^3$. The universal coefficient theorem for homology over a PID Poincaré duality for oriented topological manifolds