DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Epsilon from the filtered knot Floer complex

Definition

Work over $F_{2} = Z / 2 Z$ . Let $K \subset S^{3}$ be a smooth oriented knot and let $C = {C F K}^{\infty} (K)$ be its finitely generated free $F_{2} [U, U^{- 1}]$ chain complex with Maslov grading $M$ and bifiltration coordinates $(i, j)$ . Multiplication by $U$ lowers each filtration coordinate by one and lowers $M$ by two; the differential lowers $M$ by one and does not increase either filtration. The total homology is $F_{2} [U, U^{- 1}]$ with its generator normalized in Maslov grading zero.

For a filtration region $R$ , write $C {R}$ for the corresponding subquotient of the bifiltered complex; in particular $C {i = 0}$ is the vertical complex, $C {j = s}$ is the horizontal complex at Alexander level $s$ , and $C {i = 0, j \leq s}$ is the subcomplex of the vertical complex at levels at most $s$ . Define $τ (C) = \min {s \in Z : H_{*} (C {i = 0, j \leq s}) ⟶ H_{*} (C {i = 0}) is nonzero} .$ For each integer $s$ , let $F_{s}$ be the homology map induced by the chain map from $C {i = 0}$ to $C {\min (i, j - s) = 0}$ obtained by quotienting by $C {i = 0, j < s}$ and then including the quotient. Let $G_{s}$ be the homology map from $C {\max (i, j - s) = 0}$ to $C {i = 0}$ obtained by quotienting by $C {i < 0, j = s}$ and then including. The epsilon invariant is $ϵ (C) = {\begin{cases} 1, & F_{τ (C)} = 0, \\ - 1, & G_{τ (C)} = 0, \\ 0, & F_{τ (C)} \neq 0 and G_{τ (C)} \neq 0. \end{cases}$ Here zero and nonzero refer to the induced homology maps. This definition fixes the sign convention; invariance under stable equivalence is a separate claim.

Definition references

The diagram-specific knot Floer filtration and tau convention

Hom, The knot Floer complex and the smooth concordance group, §§2–3

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

Facts & Assumptions

Given: A finitely generated bifiltered knot Floer complex $C$ with the grading, filtration, and rank-one vertical homology conventions stated above.

[A1]

The Alexander filtration and tau level: that item defines the threshold for specified normalized filtered hat data and does not itself prove diagram/surface independence or state the Laurent subquotient interface used here. The current Grid homology is independent of the grid and computes knot Floer homology separately supplies a normalized grid-to-analytic filtered comparison; localization gives the Laurent comparison under its stated prerequisites. This Definition remains an algebraic definition on a supplied complex.

[F1]

The two bifiltered hook maps cannot both kill the threshold class proves that a vertical class represented at $j \leq s$ and killed by $F_{s}$ is in the image of $G_{s}$ , by explicit northeast/southwest hook projections. It uses no simultaneous horizontal simplification theorem.

Proof

1.1givenF1algebra

The quotient and inclusion constructions defining $F_{s}, G_{s}$ are chain maps between the stated subquotient complexes, since the differential cannot increase either coordinate. The vertical complex is finite-dimensional: translate each of the finitely many homogeneous Laurent basis generators to algebraic level zero. Its remaining basis has finitely many integer Alexander levels. Since its homology is one-dimensional by the Given data, some finite cycle represents its nonzero class. Below the least basis level the filtered subcomplex is zero, and above the greatest level it is the whole vertical complex. Thus the minimum defining $τ (C)$ exists.

2.1givenF1step 1.1algebra∎

Let $s = τ (C)$ and let $ξ$ be the nonzero vertical homology class, with a cycle representative supported at $j \leq s$ . If $F_{s} = 0$ , [F1] produces a southwest-hook cycle whose image under $G_{s}$ is $ξ$ , so $G_{s} \neq 0$ . Hence the two maps cannot both vanish. Their remaining zero/nonzero possibilities are exactly the three displayed cases, proving the algebraic trichotomy and the value set ${- 1, 0, 1}$ . This argument assumes no horizontal-representative equivalence, and supplies no model independence.

Unresolved proof obligations

The diagramwise tau definition alone does not supply an analytic ${C F K}^{\infty}$ model. The current Grid homology is independent of the grid and computes knot Floer homology has an authored proof of the normalized grid-to-analytic comparison; localization gives the Laurent comparison under that supplier's stated prerequisites. This Definition's two-hook proof establishes only the algebraic trichotomy for a supplied finite based bifiltered complex with rank-one vertical homology. It does not prove stable-equivalence or concordance invariance, nor the simultaneous horizontal-representative normal form supplied separately by Finite filtered splittings preserve the epsilon maps under total-acyclic stabilization.

Depends on

Used by

A low-weight basis term does not define Upsilon Counterexample
The Upsilon function from the filtered knot Floer complex Definition
Epsilon and Upsilon for the trefoil and its mirror Example
Epsilon is stable-equivalence invariant but not a homomorphism Lemma
Epsilon and Upsilon concordance data Theorem
The grid Upsilon function agrees with the original Upsilon invariant Theorem
Upsilon additivity, mirror symmetry, and four-ball genus bound Theorem

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jennifer Hom, The knot Floer complex and the smooth concordance group, §§2–3 (standard reference, not scraped)
Jennifer Hom, A survey on Heegaard Floer homology and concordance, §3.1 (standard reference, not scraped)