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

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Epsilon and Upsilon concordance data

Statement

For the knot-level assertions, assume the Axiom of Countable Choice. Let $C_{s m}$ be the smooth knot concordance group, with addition induced by oriented connected sum and inverse represented by reverse mirror. The function $ϵ : C_{s m} \to {- 1, 0, 1}$ is well-defined and concordance-invariant, but it is not a homomorphism to $Z$ . The map $Υ : C_{s m} ⟶ P L ([0, 2], R), [K] ⟼ Υ_{K},$ is a group homomorphism, where $P L ([0, 2], R)$ is the abelian group of continuous piecewise-linear real-valued functions under pointwise addition. For every representative $K$ and $0 \leq t \leq 1$ , its value satisfies $∣ Υ_{K} (t) ∣ \leq t g_{4}^{s m} (K)$ .

Statement references

Smooth concordance classes and the smooth concordance group

Stable filtered equivalence under smooth concordance

Epsilon stability, sign, tensor rules, and non-additivity

Upsilon additivity, mirror symmetry, and genus bound

Hom, The knot Floer complex and the smooth concordance group, Proposition 3.2 and §4

Ozsvath, Stipsicz, and Szabo, Concordance homomorphisms from knot Floer homology, §§1–5

Livingston, Notes on the knot concordance invariant Upsilon, §§7–12

Facts & Assumptions

Given: The smooth concordance group, the epsilon invariant, the Upsilon function, and the smooth four-ball genus with the conventions stated above. Assume the Axiom of Countable Choice for the supplied geometric knot Floer and concordance/genus interfaces; the ensuing finite deductions require no further choice.

[F1]

Smooth and topological knot concordance classes form abelian groups: in the smooth branch, the operation is oriented connected sum, the inverse is reverse mirror, and a class is zero precisely when represented by a smoothly slice knot. The smooth proof branch was inspected; the topological branch is not used.

[F2]

The Upsilon supplier proves its model invariance, finite piecewise linearity, reflection, tensor additivity, mirror sign and initial slope. Its smooth four-ball genus clause uses the explicitly external OSS/Hom–Wu nu-minus bound recorded there. Upsilon additivity, mirror symmetry, and four-ball genus bound

[F3]

The completed epsilon supplier gives its actual stable-core/hook trichotomy, stable-equivalence invariance, normalized conjugation/dual sign and tensor rules, and admitted trefoil and torus/cable witnesses. In particular the trefoil and its double both have epsilon1, proving integer nonadditivity. This does not itself prove smooth-concordance descent. Epsilon is stable-equivalence invariant but not a homomorphism

[F4]

Smooth concordance implies stable filtered equivalence using the explicitly external Hom slice splitting and the preserved local finite tensor-stabilization argument. Its actual normalized model and tensor interfaces are supplied. Smooth concordance implies stable filtered equivalence of knot Floer complexes

Proof

1.1givenF1F2

If $K$ and $J$ represent the same smooth concordance class, then $K # (- J)$ is smoothly slice by [F1], so $g_{4}^{s m} (K # (- J)) = 0$ . The genus bound in [F2] gives $Υ_{K # (- J)} (t) = 0$ for $0 \leq t \leq 1$ ; reflection gives the same for all $t \in [0, 2]$ .

1.2givenF3F4

If $K$ and $J$ are smoothly concordant, [F4] gives stable filtered equivalence of their knot Floer complexes. Stable-equivalence invariance in [F3] then gives $ϵ (K) = ϵ (J)$ , so epsilon is well-defined on $C_{s m}$ . The trefoil witness in [F3] shows that it is not additive as a $Z$ -valued function.

2.1givenF1F2step 1.1algebra∎

By additivity and reverse-mirror antisymmetry in [F2], $0 = Υ_{K # (- J)} (t) = Υ_{K} (t) - Υ_{J} (t),$ so $Υ_{K} = Υ_{J}$ and the function is well-defined on smooth concordance classes. The same additivity identity on arbitrary representatives shows that $[K] \mapsto Υ_{K}$ is a homomorphism to $P L ([0, 2], R)$ .

External-input boundary

The exact epsilon algebra and admitted nonadditivity witness are preserved local inputs. Concordance descent uses [F4]'s explicitly external Hom slice splitting; the Upsilon genus clause uses [F2]'s explicitly external OSS/Hom–Wu bound. The remaining group and finite minimax deductions are the displayed local argument. No full-AC assumption is inferred from ACω, and no new local movie-map construction is claimed.

Depends on

Used by

A low-weight basis term does not define Upsilon Counterexample
Epsilon and Upsilon for the trefoil and its mirror Example

Dependency tree · two levels

57 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, Proposition 3.2 and §4 (standard reference, not scraped)
Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo, Concordance homomorphisms from knot Floer homology, §§1–5 (standard reference, not scraped)
Charles Livingston, Notes on the knot concordance invariant Upsilon, §§7–12 (standard reference, not scraped)