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A low-weight basis term does not define Upsilon

Statement refuted

Let $C = {C F K}^{\infty} (T_{2, 3})$ be the right-handed trefoil staircase on generators $x_{0}, x_{1}, x_{2}$ with $(M, A) (x_{0}) = (0, 1)$ , $(M, A) (x_{1}) = (- 1, 0)$ , $(M, A) (x_{2}) = (- 2, - 1)$ and differential $\partial x_{1} = U x_{0} + x_{2}$ , $\partial x_{0} = \partial x_{2} = 0$ . The claim that $Υ_{T_{2, 3}} (1)$ is obtained by minimizing $F_{1} (x)$ over the individual displayed basis terms $x_{0}, x_{1}, x_{2}$ , instead of over Maslov-zero cycles representing the normalized generator, is false. Assuming the Axiom of Countable Choice for the normalized grid/analytic comparison, under the diagonal-grid convention every grid diagram of the trefoil also has grid Upsilon value $- 1$ at $t = 1$ .

Statement references

The cycle-minimax definition, including the normalized-generator condition

Ozsvath, Stipsicz, and Zoltan Szabo, Concordance homomorphisms from knot Floer homology, §§2 and 6

Livingston, Notes on the knot concordance invariant Upsilon, §§4–6

Facts & Assumptions

Given: The explicitly displayed right-handed trefoil staircase and the definition of $F_{1}$ on a finite chain.

[F1]

The Upsilon function from the filtered knot Floer complex: $F_{1} (z)$ is the maximum weight among terms in the support of $z$ , and $ν_{1}$ minimizes this value over Maslov-zero cycles representing the normalized generator.

[F2]

The grid Upsilon function agrees with the original Upsilon invariant now contains a proof of the grid-value comparison using the normalized grid/Heegaard supplier, whose own current Proof is present and whose [F1]–[F5] prerequisites remain explicit. MOS Proposition 2.3 supplies the chosen-variable filtered comparison and tilde-homology $V$ -factor, and Theorem 3.3 identifies the grid filtered complex with the knot filtration; the current in-run comparison proof supplies the chain-level collapse factor and its absolute normalization argument.

Counterexample

1.1givenF1algebra

The only Maslov-zero basis terms in the displayed staircase are $x_{0}$ and $U^{- 1} x_{2}$ ; $x_{1}$ has odd Maslov parity, so no integral $U$ -translate of $x_{1}$ has Maslov grading zero. Both even-degree terms are cycles, and $\partial (U^{- 1} x_{1}) = x_{0} + U^{- 1} x_{2}$ shows they represent the same nonzero generator. Their coordinates are $(0, 1)$ and $(1, 0)$ , each of $F_{1}$ -weight $1 / 2$ . The sum is a boundary, so every Maslov-zero cycle representing the normalized generator has filtration $1 / 2$ . Thus $ν_{1} = 1 / 2$ and $Υ (1) = - 2 (1 / 2) = - 1$ .

2.1givenF1step 1.1algebra

The displayed basis term $x_{2}$ has coordinates $(0, - 1)$ and weight $F_{1} (x_{2}) = - 1 / 2$ , which is less than $1 / 2$ . It has Maslov grading $- 2$ , so it is excluded from the Maslov-zero representative set; although it is a cycle, its homology class is not the normalized degree-zero generator. The false rule would return $- 2 (- 1 / 2) = + 1$ , contradicting step 1.1.

3.1givenF2step 1.1algebra∎

Under Countable Choice, [F2] gives that the diagonal grid complex has the same Upsilon value as the original invariant, so step 1.1 gives the stated grid value $- 1$ . The grid comparison uses the supplier prerequisites recorded in [F2].

Unresolved proof obligations

The finite staircase refutation is complete. Its right-handed trefoil identification is supplied by the signed three-crossing grid proof in Epsilon is stable-equivalence invariant but not a homomorphism, step 1.4, together with that item's finite staircase computation. The universal grid-value clause is supplied by [F2] under the comparison prerequisites it records; the earlier missing-comparison escalation is no longer current.

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Sources

Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo, Concordance homomorphisms from knot Floer homology, §§2 and 6 (standard reference, not scraped)
Charles Livingston, Notes on the knot concordance invariant Upsilon, §§4–6 (standard reference, not scraped)