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

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The leading minus sign in the Dubrovnik conversion is necessary

Statement refuted

The formula obtained by omitting the leading minus sign, $D_{K} (a, z) = i^{- w (K)} (- 1)^{c (K)} L_{K} (i a, - i z), i^{2} = - 1,$ is false: on the one-component unknot it gives $D = - 1$ , while the declared Dubrovnik normalization gives $D = 1$ .

Statement references

Unknot normalization and conversion convention

Kauffman, An Invariant of Regular Isotopy (1990), §VII

Diagram

The circle below depicts the one-component test link; the numerical refutation follows from its declared writhe, component count, and normalization.

Facts & Assumptions

Given: The one-component unknot, its writhe and component count, and the declared Dubrovnik and Kauffman values.

[F1]

The unknot has $w = 0$ , $c = 1$ , $D = 1$ , and $L = 1$ under the stated normalization (Kauffman regular-isotopy and Dubrovnik conventions).

Refutation

1.1F1algebra

Apply the proposed false formula to the unknot from [F1]. Since $i^{- w} = i^{0} = 1$ and $(- 1)^{c} = (- 1)^{1} = - 1$ , it assigns $D = 1 \cdot (- 1) \cdot 1 = - 1$ . This contradicts the declared value $D = 1$ in [F1].

2.1F1step 1.1algebra∎

Inserting the leading minus changes that result to $- 1 \cdot (- 1) = 1$ , exactly the normalized unknot value from [F1]. Thus the leading sign is necessary for the displayed conversion.

Depends on

Kauffman regular-isotopy and Dubrovnik conventions

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

Louis H. Kauffman, An Invariant of Regular Isotopy, Transactions of the AMS 318(2) (1990), §VII (standard reference, not scraped)
Frederick M. Goodman and Holly Hauschild, Affine Birman–Wenzl–Murakami Algebras and Tangles in the Solid Torus, §§2.3 and 5.1–5.5 (standard reference, not scraped)