CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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A graded Euler characteristic does not determine total rank

Statement refuted

The false assertion is that a finite bigraded cochain complex over $Q$ is determined in total dimension by its graded Euler characteristic $\sum_{i, j} (- 1)^{i} \dim_{Q} H^{i, j} q^{j}$ .

Source notes

No external statement is cited; the counterexample is finite-dimensional linear algebra.

Facts & Assumptions

Given: A finite bigraded cochain complex over $Q$ with zero differential. Its total rank is the sum of the dimensions of all bigraded groups, and its graded Euler characteristic weights degree $(i, j)$ by $(- 1)^{i} q^{j}$ .

[F1]

With zero differential, every cohomology group equals its corresponding cochain group by the definition of cohomology.

Counterexample

1.1givenconstruct

Define $V$ by $V^{0, 0} = Q$ and all other groups zero. Define $W$ by $W^{0, 0} = W^{0, 1} = W^{1, 1} = Q$ and all other groups zero; give both complexes zero differentials.

2.1F1step 1.1algebra∎

By [F1], the graded Euler characteristics are $χ_{q} (V) = 1$ and $χ_{q} (W) = 1 + q - q = 1$ , but their total dimensions are $1$ and $3$ . Therefore equal graded Euler characteristics do not determine total rank.

Remarks

This vector-space example does not assert that a nontrivial knot has the unknot's Jones polynomial. Jones-polynomial unknot detection is a separate knot-theoretic problem; a 2026 survey still describes it as open.

Remarks references

Manolescu, From Knots to Four-Manifolds (2026), §2, p. 5

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Ciprian Manolescu, From Knots to Four-Manifolds, §2 (standard reference, not scraped)