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The filtered knot Floer complex of a connected sum is a tensor product

Statement

Assume the Axiom of Countable Choice. Let $K$ and $J$ be smooth oriented knots in the standard oriented three-sphere, with chosen minimum-genus oriented Seifert surfaces $F_{K}$ and $F_{J}$ . Use the boundary-connected-sum surface $F_{K} ♮ F_{J}$ to normalize the connected-sum model. A normalized $C F K^{\infty} (K)$ is a free finite-rank chain complex over $F_{2} [U, U^{- 1}]$ with two integer filtrations $(i, j)$ , where multiplication by $U$ lowers both coordinates by one. Give the connected sum $K # J$ the fixed oriented gluing convention and the sum filtrations and Maslov/Alexander gradings. Then $C F K^{\infty} (K # J)$ is bifiltered chain homotopy equivalent to $C F K^{\infty} (K) \otimes_{F_{2} [U, U^{- 1}]} C F K^{\infty} (J)$ , with an equivalence and homotopies preserving both filtration coordinates. Moreover, the normalized finite filtered hat complex of $K # J$ is filtered chain homotopy equivalent to $C_{K} \otimes C_{J}$ with tensor filtration $F_{s} (C_{K} \otimes C_{J}) = \sum_{p + q \leq s} F_{p} C_{K} \otimes F_{q} C_{J}$ . Consequently, $τ (K # J) = τ (K) + τ (J) .$

Statement references

The normalized finite filtered complex and tau threshold

The fixed oriented connected-sum gluing convention

Additivity of the threshold under a filtered tensor product

Independence of local connected-sum choices

Ozsvath and Szabo, Knot Floer homology and the four-ball genus, Proposition 3.2

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

Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links, Chapters 5, 6, and 13

Facts & Assumptions

Given: Countable choice, the knots and their chosen oriented Seifert surfaces, and the fixed oriented connected-sum convention.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is the sequence-choice principle assumed in the Statement.

[F1]

The Alexander filtration and tau level defines the finite normalized filtered hat complex and its threshold for the specified diagram and Seifert surface.

[F2]

Oriented connected sum and prime knot fixes the oriented connected-sum convention, and The oriented connected sum is independent of its local choices identifies the resulting knot type independently of the local gluing choices.

[F3]

The tau level adds under the filtered tensor product proves additivity of the threshold for the finite tensor filtration when both total homology groups are one-dimensional.

[F4]

External input, authorized by the run-specific source boundary. Ozsváth and Szabó, Holomorphic disks and knot invariants, §7, Theorem 7.1, PDF pp. 43–45, states filtered chain homotopy equivalences for the knot complexes $C F K^{0}$ , $C F K^{-}$ , and $C F K^{\infty}$ under connected sum, using the boundary-connected-sum Seifert surface and the sum of the Alexander and Maslov gradings. The hypotheses apply to knots in $S^{3}$ . The theorem is stated over $Z$ ; tensoring its chain maps, inverse maps, and homotopies with $F_{2}$ preserves their identities. Its tensor filtration on homogeneous generators gives the convolution filtration displayed in the Statement. I read the theorem and its proof sketch; the proof constructs the standard triangle map and refers to Ozsváth–Szabó, Holomorphic disks and three-manifold invariants: properties and applications, §6, for the ordinary connected-sum map. I read that section through Theorem 6.2, but do not claim to reconstruct the knot-filtration adaptation stated in Theorem 7.1.

[F5]

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 the standard $τ$ invariant is a homomorphism from the smooth knot concordance group to $Z$ . Hom's threshold definition in §3.1 is the same least filtered level as [F1]; its target is $\hat{H F} (S^{3}) ≅ F_{2}$ , so “nonzero image” and “surjective” agree. The survey states on printed p. 2 that it works throughout with $F_{2}$ coefficients. I read the definition and theorem statement; I do not claim to reconstruct the theorem's cited proof.

Proof

Given: The hypotheses of the Statement.

1.1F1F2F4givenconstruct

Apply the external connected-sum theorem [F4] to the chosen doubly pointed models for $(K, F_{K})$ and $(J, F_{J})$ . Its $C F K^{\infty}$ conclusion gives the required bifiltered chain homotopy equivalence, and its $C F K^{0}$ conclusion gives the finite hat equivalence for the boundary-connected-sum surface. The theorem's maps and inverse homotopies are over the stated coefficient ring; base change from $Z$ to $F_{2}$ preserves each chain-map and homotopy identity. The source's generator filtration is the sum of the input Alexander levels, so the induced increasing filtration is exactly $\sum_{p + q \leq s} F_{p} C_{K} \otimes F_{q} C_{J}$ .

2.1F1step 1.1algebra

A filtered chain homotopy equivalence between finite complexes with one-dimensional total homology preserves the threshold. Indeed, each filtered map commutes with the inclusion into total homology, and the induced total map is an isomorphism; a nonzero class from one filtered subcomplex therefore maps to a nonzero class from the corresponding subcomplex of the other. Applying this in both directions shows that the least such filtration level agrees.

3.1F3F5step 1.1step 2.1algebra∎

The two total homology groups are $F_{2}$ , so their tensor has one-dimensional total homology. Apply the finite tensor-threshold formula [F3] to the equivalence in step 1.1, obtaining additivity for the specified diagramwise thresholds. Hom's external theorem [F5] identifies the unqualified $τ (K)$ with the common standard knot invariant and states its additivity under connected sum. This gives the final invariant-level formula without treating any unverified local analytic factorization as a premise.

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Sources

Ozsvath and Szabo, Holomorphic disks and knot invariants (standard reference, not scraped)
Ozsvath and Szabo, Holomorphic disks and three-manifold invariants: properties and applications (standard reference, not scraped)
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)