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

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Reduced and unreduced Khovanov Euler characteristics and the Jones polynomial

Statement

Assume the Axiom of Choice. Let $V_{L} (t)$ be the library Jones polynomial of a nonempty oriented link $L$ , normalized by $V_{◯} (t) = 1$ . For every diagram $D$ of $L$ , put $q = - A^{- 2}$ and $t = q^{2} = A^{- 4}$ . Then the integral rank Euler characteristics satisfy $\sum_{i, j} (- 1)^{i} {rank}_{Z} K h^{i, j} (L) q^{j} = (q + q^{- 1}) V_{L} (t) .$ If $L = K$ is a knot with any basepoint $p$ away from the crossings, then $\sum_{i, \tilde{j}} (- 1)^{i} {rank}_{Z} {\tilde{K h}}^{i, \tilde{j}} (K, p) q^{\tilde{j}} = V_{K} (t) .$ The unreduced factor $q + q^{- 1}$ and reduced factor $1$ are the respective unknot normalizations.

Statement references

Full Axiom of Choice

Full choice implies countable choice

Statewise Euler characteristic and bracket normalization

Bracket-to-library Jones comparison

Unreduced link invariance

Reduced basepoint and diagram invariance

Khovanov, A categorification of the Jones polynomial (1999), §§2.4 and 7.1–7.2

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

Facts & Assumptions

Given: Full AC, a nonempty oriented link $L$ , its finite-diagram complexes, and the library Jones polynomial.

[A1]

Full AC means every family of nonempty sets has a choice function (The Axiom of Choice).

[F1]

Full AC implies AC $_{ω}$ (AC implies DC implies countable choice).

[F2]

For a fixed finite diagram $D$ , the integral rank Euler sums of its Khovanov cohomology are $(q + q^{- 1}) f_{D} (A)$ unreduced and $f_{D} (A)$ reduced, where $q = - A^{- 2}$ and $f_{D}$ is the writhe-normalized state sum (The graded Euler characteristics have the Jones normalization).

[F3]

Under full AC, the normalized bracket state sum equals the library Jones polynomial at $t = A^{- 4}$ (The bracket state sum matches the library Jones specialization).

[F4]

Under AC $_{ω}$ , the unreduced groups are independent of the diagram, and the reduced groups of a knot are independent of the diagram and basepoint (Khovanov homology is an oriented link invariant, Reduced Khovanov cohomology is invariant under based moves and basepoint transport).

Proof

1.1A1F1F2F4givenalgebra

By [A1] and [F1], full AC supplies AC $_{ω}$ . Hence [F4] identifies the bigraded groups computed from any finite diagram $D$ with $K h^{i, j} (L)$ , and in the knot case identifies the reduced groups with ${\tilde{K h}}^{i, \tilde{j}} (K, p)$ for every allowed basepoint. The fixed-diagram complexes have finite rank and bounded support, so their rank Euler sums are finite.

2.1F2F3F4step 1.1algebra∎

By [F2], those Euler sums are $(q + q^{- 1}) f_{D} (A)$ and $f_{D} (A)$ , respectively. By [F3], $f_{D} (A) = V_{L} (A^{- 4})$ under the stated full-AC hypothesis. Substituting $q = - A^{- 2}$ and $t = q^{2} = A^{- 4}$ gives exactly the two formulas in the Statement. For a crossingless unknot the unreduced complex has its two generators in quantum degrees $1, - 1$ , whereas the reduced interval module has one generator in reduced degree zero, so the respective factors are $q + q^{- 1}$ and $1$ . Since [F4] makes the groups diagram- and basepoint-independent, these are the claimed link and knot Euler characteristics.

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Sources

Mikhail Khovanov, A categorification of the Jones polynomial, arXiv:math/9908171v2, §§2.4 and 7.1–7.2 (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 (standard reference, not scraped)