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

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The Kauffman and Dubrovnik polynomials agree under the stated conversion

Statement

Assume the Axiom of Choice. Fix any nonempty oriented regular link diagram $K$ , with determinant writhe $w (K)$ and component count $c (K)$ . Let $Q_{K} (A, Z) \in T_{K}$ denote the value returned by the BMW braid-closure algorithm for any finite signed braid word whose closure represents the oriented link represented by $K$ . The quantities $D_{K}$ and $L_{K}$ below are evaluated on this fixed diagram; $F_{K}$ and $Q_{K}$ depend only on its ambient oriented link type. Then $Q_{K}$ is independent of the chosen braid and $Q_{K} (A, Z) = F_{K} (A, Z) = - i^{w (K)} (- 1)^{c (K)} A^{- w (K)} D_{K} (- i A, i Z),$ with the inverse conversion $D_{K} (a, z) = - i^{- w (K)} (- 1)^{c (K)} L_{K} (i a, - i z), i^{2} = - 1.$ The proof covers the unknot, positive and negative curls, and both self-crossing split and mixed-crossing merge smoothing cases under the stated universal coefficient relation, without dividing by $z$ .

Statement references

Kauffman and Dubrovnik values

Local writhe, component, and skein conversion

BMW braid-closure algorithm

Termination and output formula for the procedure

Markov trace invariance across braid representatives

Kauffman, An Invariant of Regular Isotopy

Goodman and Hauschild, Affine Birman–Wenzl–Murakami Algebras and Tangles in the Solid Torus

Birman and Brendle, Braids: A Survey

Facts & Assumptions

Given: AC, a fixed nonempty oriented regular link diagram, the BMW trace computation from a braid representative, and the declared Dubrovnik/Kauffman conventions.

[A1]

The Axiom of Choice, used for the closed-braid representation and Markov equivalence theorems (The Axiom of Choice).

[F1]

The BMW trace output is independent of the braid representative and equals the declared Kauffman value $F (i a, - i z)$ (The BMW Markov trace gives the Dubrovnik link invariant).

[F2]

For every oriented diagram with writhe $w$ and component count $c$ , $D = - i^{- w} (- 1)^{c} L (i a, - i z)$ ; the proof of that result verifies both skeins, curls, and orientation independence (Kauffman and Dubrovnik writhe-component conversion).

[F3]

The scalar Dubrovnik evaluation has unknot value $D = 1$ , positive and negative curl factors $a, a^{- 1}$ , and satisfies the coefficient relation $z (d - 1) = a - a^{- 1}$ (Kauffman regular-isotopy and Dubrovnik conventions, Scalar evaluation of framed Kauffman link diagrams).

[F4]

On each admissible finite signed braid word, the procedure terminates and returns its corrected BMW trace expression (The BMW braid-closure algorithm terminates and returns its stated value).

Proof

1.1A1F1F4given

Since $K$ is nonempty, [A1] and Alexander's theorem give a finite braid representative. Write $Q^{o l d} (β) = - (- 1)^{c} a^{- w} d^{n - 1} τ_{n} (β)$ . By [F4], the algorithm returns $φ (Q^{o l d} (β))$ : indeed $φ (a^{- w}) = i^{w} A^{- w}$ . By [F1], $Q^{o l d} (β)$ is independent of the braid representative and equals $F_{K} (i a, - i z)$ . Applying the ring homomorphism $φ$ , with $a = - i A, z = i Z$ , gives $Q_{K} (A, Z) = F_{K} (A, Z)$ and preserves independence.

2.1F2step 1.1algebra

By [F2], $L_{K} (i a, - i z) = - i^{w} (- 1)^{c} D_{K} (a, z)$ . Substitute $a = - i A, z = i Z$ and multiply by $A^{- w}$ to obtain $F_{K} (A, Z) = - i^{w} (- 1)^{c} A^{- w} D_{K} (- i A, i Z) .$ Together with step 1.1 this is the stated algorithm identity. The unsubstituted [F2] is exactly the displayed inverse conversion.

3.1F2F3step 2.1algebra∎

The boundary cases agree with this formula: for the unknot, $c = 1, w = 0, D = 1$ , so $Q_{K} (A, Z) = 1$ ; a positive curl changes $(w, D)$ to $(w + 1, a D)$ , leaving $a^{- w} D$ unchanged, and a negative curl changes them to $(w - 1, a^{- 1} D)$ with the same result. After substitution, the positive-curl multiplier in $i^{w} A^{- w} D_{K} (- i A, i Z)$ is $i A^{- 1} (- i A) = 1$ , and the negative-curl multiplier is $i^{- 1} A (- i A)^{- 1} = 1$ . At a self-crossing, the oriented smoothing splits one component and the other smoothing keeps the component count. At a mixed crossing, both smoothing pairings merge the two incident components; orienting the non-oriented smoothing by reversing one incident whole component gives the modulo-four writhe phase established from the even mixed-crossing sums in [F2]. Thus the sum/difference skein conversions cover both local component changes, and no coefficient is divided by $z$ .

Depends on

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Sources

Louis H. Kauffman, An Invariant of Regular Isotopy, Transactions of the AMS 318(2) (1990), §§VI–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)
Joan S. Birman and Tara E. Brendle, Braids: A Survey, Handbook of Knot Theory chapter, Theorems 2 and 4 (standard reference, not scraped)