CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A birth and a death do not cancel

Statement refuted

The assertion that a birth of a disjoint circle followed by a death of that same circle induces the identity up to an overall sign on an unaffected circle is false. Let $A = Z [X] / (X^{2})$ with unit $η (1) = 1$ and counit $ϵ (1) = 0, ϵ (X) = 1$ . On a movie with one unaffected circle, birth a disjoint circle and then cap that same circle by a death. The composite operator is

A \overset{{id}_{A} \otimes η}{\to} A \otimes A \overset{{id}_{A} \otimes ϵ}{\to} A,

which is zero and therefore is not the identity, even after allowing multiplication by an overall sign.

diagram failed to render:
\begin{tikzcd}
\mathcal A
\arrow[r, "{\operatorname{id}\otimes\eta}"]
& \mathcal A\otimes\mathcal A
\arrow[r, "{\operatorname{id}\otimes\epsilon}"]
& \mathcal A
\end{tikzcd}

The movie is a cylinder on the original circle disjoint union a born-and-capped sphere; the diagram specifies the local birth and death sequence and is not used as a proof.

Statement references

The unit, counit, and integral Frobenius algebra

Khovanov, A categorification of the Jones polynomial, §7

Bar-Natan, Khovanov's homology for tangles and cobordisms, §§7–8

Kauffman, An Introduction to Khovanov Homology, §5

Facts & Assumptions

Given: The free abelian group $A = Z [X] / (X^{2})$ , with $η (1) = 1$ , $ϵ (1) = 0$ , and $ϵ (X) = 1$ .

[F1]

Khovanov Frobenius algebra and cube of resolutions gives the unit and counit formulas and the nonzero basis element $1 \in A$ .

Counterexample

1.1F1algebra

The birth map sends $1 \in Z$ to $1 \in A$ , and the following death sends this element to $ϵ (1) = 0$ . Thus $ϵ η = 0 : Z \to Z$ .

2.1F1step 1.1algebra∎

Tensoring the zero map $ϵ η$ with ${id}_{A}$ gives the zero endomorphism of $A$ , which is the composite in the statement. The identity of $A$ is nonzero because it sends $1$ to $1 \neq 0$ ; its negative also sends $1$ to $- 1 \neq 0$ . Hence the composite is neither the identity nor its negative.

Depends on

Khovanov Frobenius algebra and cube of resolutions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Mikhail Khovanov, A categorification of the Jones polynomial, §7 (standard reference, not scraped)
Dror Bar-Natan, Khovanov's homology for tangles and cobordisms, §§7–8 (standard reference, not scraped)
Louis H. Kauffman, An Introduction to Khovanov Homology, §5 (standard reference, not scraped)