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

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Tau is an additive smooth concordance invariant and bounds four-ball genus

Statement

Assume the Axiom of Countable Choice. For every smooth oriented knot $K$ in the standard oriented three-sphere, the common normalized filtered knot Floer threshold $τ (K)$ is independent of the admissible diagram and is invariant under smooth concordance. For all smooth oriented knots $K, J$ , it satisfies $τ (K # J) = τ (K) + τ (J) .$ If $g_{4}^{s m} (K)$ is the minimum genus of a smooth compact connected oriented properly embedded surface in the four-ball with boundary $K$ , then $∣ τ (K) ∣ \leq g_{4}^{s m} (K) .$

Statement references

The normalized filtered threshold defining tau

Smooth concordance with product collars

The smooth four-ball genus

The diagram and grid comparison needed for a knot invariant

The genus shift for smooth knot cobordisms

The filtered tensor product for connected sums

Ozsvath and Szabo, Knot Floer homology and the four-ball genus, Theorem 1.1

Hom, A survey on Heegaard Floer homology and concordance, §§3.1–3.2

Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links, Chapters 6, 8, and 13

Facts & Assumptions

Given: The Axiom of Countable Choice, smooth oriented knots, and the standard normalized filtered knot Floer models.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) states the axiom retained in the Statement.

[F1]

The Alexander filtration and tau level defines the diagramwise threshold as the least Alexander level whose inclusion has nonzero image in total homology. Hom's §3.1 definition uses the same normalized knot Floer filtration; because $\hat{H F} (S^{3}) ≅ F_{2}$ , nonzero image is equivalent to surjectivity.

[F2]

External input, authorized by the run-specific source boundary. Hom, A survey on Heegaard Floer homology and concordance, §3.1, Theorem 3.1(1), printed p. 8, states that $τ$ is a surjective homomorphism from the smooth knot concordance group to $Z$ . It therefore includes diagram independence, smooth concordance invariance, and connected-sum additivity. The survey states on printed p. 2 that it works throughout with $F_{2}$ coefficients. I read the threshold definition and theorem statement; I do not claim to reconstruct the cited proof.

[F3]

External input, authorized by the run-specific source boundary. Hom, §3.1, Theorem 3.1(2), printed p. 8, states $∣ τ (K) ∣ \leq g_{4} (K)$ for the smooth four-ball genus.

[F4]

Smooth and topological slice genus uses the same minimum over smooth connected oriented properly embedded surfaces in $B^{4}$ as Hom's $g_{4}$ .

Proof

Given: The hypotheses of the Statement.

1.1F1givenalgebra

The threshold in [F1] is Hom's standard $τ$ definition: both use the normalized Alexander-filtered knot Floer complex and the first level whose inclusion maps nontrivially to total homology. The target is one-dimensional, so the nonzero-image and surjectivity formulations agree. Thus Hom's $τ (K)$ is the common normalized threshold asserted here.

2.1F2step 1.1algebra

Hom's homomorphism theorem [F2] gives diagram independence, invariance under smooth concordance, and $τ (K # J) = τ (K) + τ (J)$ .

3.1F3F4step 1.1algebra∎

Hom's genus theorem [F3], with the identical genus convention [F4], gives $∣ τ (K) ∣ \leq g_{4}^{s m} (K)$ . This proves the remaining conclusion.

Depends on

Used by

Dependency tree · two levels

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Sources

Peter S. Ozsvath and Zoltan Szabo, Knot Floer homology and the four-ball genus (standard reference, not scraped)
Jennifer Hom, A survey on Heegaard Floer homology and concordance (standard reference, not scraped)
Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo, Grid Homology for Knots and Links (standard reference, not scraped)