ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Unknot Khovanov and reduced Khovanov groups

Example

Let $U$ be the crossingless diagram of the oriented unknot, and choose any basepoint $p$ on its unique circle. For the normalized integral Khovanov cochain complex, $K h^{0, 1} (U) ≅ Z$ , $K h^{0, - 1} (U) ≅ Z$ , and all other unreduced groups vanish. For the reduced quotient convention that kills the based-circle $X$ label and lowers the quantum degree of the surviving label $1$ by one, ${\tilde{K h}}^{0, 0} (U, p) ≅ Z$ and all other reduced groups vanish. Consequently the unreduced and reduced rank Euler polynomials are $q + q^{- 1}$ and $1$ , respectively.

Example references

Integral rank-two Frobenius algebra and grading

Zero-crossing state cube

Based quotient and normalized reduced grading

Unreduced and reduced Euler normalizations

Khovanov, A categorification of the Jones polynomial (1999), §§2.2, 4.2, and 7.2

Bar-Natan, On Khovanov's categorification of the Jones polynomial (2002), §§3.1 and 3.5

Kauffman, An Introduction to Khovanov Homology (2022), §§3–4

Zhang, Notes on Khovanov homology (2025), §§3.5–3.7 and 6.3.4

Facts & Assumptions

Given: The crossingless one-circle diagram $U$ and a basepoint $p$ on that circle.

[F1]

The crossing set is empty, so there is one state and its state module is one copy of $A$ in cohomological degree $0$ ; there are no edge maps (Khovanov chain complex from a diagram).

[F2]

The basis of $A = Z [X] / (X^{2})$ is $1, X$ with quantum degrees $1, - 1$ (Khovanov Frobenius algebra and cube of resolutions).

[F3]

The reduced quotient by the image of the basepoint action removes $X$ and shifts the surviving $1$ generator's quantum degree from $1$ to $0$ (Reduced Khovanov cohomology of a knot).

Verification

1.1F1F2givenalgebra

By [F1], the zero-crossing cube has just $C^{0} (U) = A$ and differential zero. Therefore its cohomology equals $A$ in cohomological degree $0$ . By [F2], the two basis elements give one free rank-one group in quantum degree $1$ and one in degree $- 1$ , with no other groups.

2.1F2F3step 1.1algebra∎

In the based quotient, the image of multiplication by $X$ is $Z {X}$ , so the quotient of $A = Z {1, X}$ is freely generated by $[1]$ . By [F3] its reduced quantum degree is $1 - 1 = 0$ ; the differential remains zero, so its only cohomology is $Z$ in bidegree $(0, 0)$ . Summing the unreduced ranks with their quantum powers gives $q + q^{- 1}$ , and the reduced rank sum is $1$ .

Depends on

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Sources

Mikhail Khovanov, A categorification of the Jones polynomial, arXiv:math/9908171v2, §§2.2, 4.2, and 7.2 (standard reference, not scraped)
Dror Bar-Natan, On Khovanov's categorification of the Jones polynomial, Algebraic & Geometric Topology 2 (2002), §§3.1 and 3.5 (standard reference, not scraped)
Louis H. Kauffman, An Introduction to Khovanov Homology, arXiv:1107.1524v2 (2022), §§3–4 (standard reference, not scraped)
Melissa Zhang, Notes on Khovanov homology, arXiv:2501.03115v1, §§3.5–3.7 and 6.3.4 (standard reference, not scraped)