LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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A slice disk induces a metabolizer on the branched-cover linking form

Statement

Assume AC. If a smooth oriented knot $K$ bounds a smooth proper disk $D$ in $B^{4}$ and $q$ is a prime power, then $P_{D} = \ker (H_{1} (Σ_{q} (K); Z) \to H_{1} (W_{D, q}; Z))$ is a metabolizer for the torsion linking form of $Σ_{q} (K)$ .

Statement references

The branched cover of a slice disk

The branched cover is a rational homology ball

The boundary-kernel metabolizer lemma

The metabolizer convention

The smooth slice-disk convention

Charles Livingston and Swatee Naik, Introduction to Knot Concordance, §8.2, pp. 167–168

Axiom of Choice

Facts & Assumptions

Given: Full AC, a smooth proper slice disk $D$ for $K$ , and a prime-power cover degree $q = p^{k}$ .

[A1]

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

[F1]

The branched cover $W_{D, q}$ has boundary $Σ_{q} (K)$ by the disk-cover convention, and it is a compact connected rational homology ball for prime-power $q$ . Cyclic branched cover of a smooth proper disk A prime-power branched cover of a smooth proper disk is a rational homology ball

[F2]

For any compact connected oriented rational homology ball with connected boundary, the kernel of boundary inclusion on $H_{1}$ is a metabolizer for the boundary linking form. The boundary kernel of a rational homology ball is a linking metabolizer

[F3]

The linking form on the finite first homology of a rational homology sphere is symmetric and nonsingular. The torsion linking pairing of a rational homology sphere is nonsingular

[F4]

A smooth knot is smoothly slice when it bounds a smooth properly embedded disk in $B^{4}$ . Smooth and topological slice genus

[F5]

A metabolizer is a subgroup equal to its orthogonal complement for the finite nonsingular linking form. Finite nonsingular linking forms and metabolizers

Proof

1.1F1F4givenA1

By [F1], the boundary of $W_{D, q}$ is the rational homology sphere $Σ_{q} (K)$ , and both are compact connected oriented manifolds with the required boundary identification. The kernel of its boundary inclusion is exactly the subgroup $P_{D}$ displayed in the statement.

2.1F2F3F5step 1.1∎

Apply [F2] to $W_{D, q}$ and its boundary. The kernel identified in step 1.1 is self-annihilating for the boundary linking form; [F3] supplies nonsingularity and [F5] the metabolizer convention. Therefore $P_{D}$ is a metabolizer.

Depends on

Used by

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Sources

Charles Livingston and Swatee Naik, Introduction to Knot Concordance (standard reference, not scraped)
Patrick Gilmer, Slice Knots in S3 (standard reference, not scraped)