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A higher cyclic branched cover can have infinite first homology

Statement refuted

Every cyclic branched cover of a smooth oriented knot has finite first homology.

Statement references

The Alexander-polynomial criterion for cyclic-cover finiteness

Every double branched cover has finite first homology

Livingston, “Seifert Forms and Concordance,” Theorem 3.1

Facts & Assumptions

Given: Full AC, the right-handed trefoil matrix $V_{T} = (\begin{array}{cc} - 1 & 0 \\ 1 & - 1 \end{array})$ , and the primitive sixth root $ζ_{6} = e^{i π / 3}$ .

[A1]

Full AC is the choice-function axiom required by the cyclic-cover and Alexander-polynomial theorems. The Axiom of Choice

[F1]

The Seifert determinant $\det (V - t V^{T})$ represents the Alexander polynomial up to a unit $\pm t^{k}$ . The Seifert determinant is the knot Alexander polynomial, The Alexander polynomial from the zeroth elementary ideal

[F2]

The cyclic-cover order theorem states that a zero at any nontrivial $q$ -th root makes the first homology infinite. Cyclic-cover homology order from the Alexander polynomial

[F3]

The Alexander normalization gives $Δ_{K} (1) = \pm 1$ , and the double-cover theorem gives finiteness when $∣ Δ_{K} (- 1) ∣$ is this nonzero odd integer. The square Seifert presentation generates the knot Alexander polynomial, Knot determinant is the order of double-cover homology

Counterexample

technique · direct

1.1F1givenalgebra

Direct calculation gives $\det (V_{T} - t V_{T}^{T}) = \det (\begin{array}{cc} t - 1 & - t \\ 1 & t - 1 \end{array}) = t^{2} - t + 1$ . By [F1], this represents the trefoil Alexander polynomial up to $\pm t^{k}$ . Since $ζ_{6}^{2} - ζ_{6} + 1 = 0$ , the Alexander polynomial vanishes at this nontrivial sixth root.

2.1A1F2F3step 1.1given∎

Applying [F2] with $q = 6$ shows that $H_{1} (Σ_{6} (T); Z)$ is infinite, contradicting the refuted universal claim. For the boundary comparison, [F3] gives $Δ_{T} (- 1)$ odd and nonzero, so $H_{1} (Σ_{2} (T); Z)$ is finite. Thus the failure occurs at a higher cyclic cover even though the double-cover case is always finite. The two supplier theorems use the Axiom of Choice in [A1].

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Sources

Charles Livingston, Seifert Forms and Concordance (standard reference, not scraped)
Shin’ichi Saeki, Knot Theory Lecture Notes, Chapter 7: Seifert Matrices and Derived Invariants (standard reference, not scraped)