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Kauffman and Dubrovnik polynomials from BMW algebras — Examples and Counterexamples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

None yet.

4 · Examples, counterexamples and false statements

CounterexampleConstruction: AI-adaptedVerification: AI-adaptedverified 2026-10-22 (gpt-6.1-sol)Open item page →

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.

ExampleConstruction: AI-adaptedVerification: AI-adaptedverified 2026-10-22 (gpt-6.1-sol)Open item page →

BMW Kauffman and Dubrovnik worked cases

Example

For the one-component unknot, the Dubrovnik value and the normalized Kauffman value are both $1$ . Adding a positive or negative curl multiplies the regular-isotopy Dubrovnik value by $a$ or $a^{- 1}$ , while the writhe correction leaves the ambient Kauffman value unchanged. For the two-component unlink, the Dubrovnik value is $d$ , the identity $2$ -braid closure has normalized trace $1$ , and the BMW algorithm returns the Kauffman value $- d$ . The Dubrovnik calculations hold in the universal coefficient ring $R_{D}$ with $z (d - 1) = a - a^{- 1}$ . The Kauffman outputs lie in the algorithm's converted ring $T_{K}$ , under $φ (a) = - i A$ , $φ (z) = i Z$ , and $φ (d) = d$ ; there $Z (d - 1) = - A - A^{- 1}$ . Neither $z$ nor $Z$ is inverted.

Example references

Polynomial and local curl/circle conventions

Normalized trace values

BMW closure algorithm

Kauffman, An Invariant of Regular Isotopy

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

Facts & Assumptions

Given: The empty closure, one- and two-strand identity braids, the normalized closure trace, and the universal Dubrovnik rules.

[F1]

On a nonempty diagram $D_{K} = P (K) / d$ , $P (◯) = d$ , a disjoint circle multiplies $P$ by $d$ , and the curl factors are $a, a^{- 1}$ (Kauffman regular-isotopy and Dubrovnik conventions, Scalar evaluation of framed Kauffman link diagrams).

[F2]

$τ_{n} (x) = d^{- n} P (\hat{Φ_{n} (x)})$ , with $a = ρ$ and $d = δ$ (BMW closure trace and stabilization scalars).

[F3]

For a braid closure with writhe $w$ and component count $c$ , the algorithm returns $- i^{w} (- 1)^{c} A^{- w} φ (d^{n - 1} τ_{n} (β))$ in $T_{K}$ , where $φ (a, z, d, q) = (- i A, i Z, d, q)$ (Compute the Kauffman polynomial from a BMW braid closure).

[F4]

The universal coefficient relation is $z (d - 1) = a - a^{- 1}$ and is not solved by dividing by $z$ (BMW relations in the Dubrovnik parameter convention).

[F6]

On each fixed oriented diagram, $L (A, Z) = - i^{w} (- 1)^{c} D (- i A, i Z)$ and $F (A, Z) = A^{- w} L (A, Z)$ are the defining formulas; the curl factors below are derived from these formulas and [F1] (Kauffman regular-isotopy and Dubrovnik conventions).

[F5]

The two-component unlink is represented by two disjoint circles; the schematic draws its two components without crossings.

[F7]

The identity $n$ -braid closes to the $n$ -component unlink, with writhe $0$ and one closure component for each fixed-point cycle; the empty braid word has exponent sum $0$ (The closure of a geometric braid, The exponent sum of a braid).

Diagram

The two disjoint circles represent the two-component unlink in [F5].

Proof

1.1F1F2F3F6F7algebra

The unknot has $P (◯) = d$ , so [F1] gives $D_{◯} = P / d = 1$ . Its identity $1$ -braid closure has $n = 1$ , $w = 0$ , $c = 1$ by [F7], and $τ_{1} (1) = d^{- 1} P (◯) = 1$ by [F2]. Substitution in [F3] gives $- (- 1)^{1} = 1$ . By [F6], $L_{◯} (A, Z) = - i^{0} (- 1)^{1} D_{◯} (- i A, i Z) = 1$ , so $F_{◯} (A, Z) = 1$ , agreeing with the algorithm output.

2.1F1F3F6step 1.1algebra

A positive curl multiplies $D$ by $a$ and raises $w$ by one, so the product $a^{- w} D$ is unchanged; a negative curl multiplies $D$ by $a^{- 1}$ and lowers $w$ by one, with the same cancellation. To obtain the corresponding Kauffman factors directly from [F6], substitute $a_{D} = - i A$ and $z_{D} = i Z$ in $D$ . A positive curl preserves $c$ , raises $w$ by one, and multiplies $D$ by $a_{D}$ , so it multiplies $L$ by $i a_{D} = A$ . A negative curl lowers $w$ by one and multiplies $D$ by $a_{D}^{- 1}$ , so it multiplies $L$ by $i^{- 1} a_{D}^{- 1} = A^{- 1}$ . Thus $A^{- (w + 1)} A L = A^{- w} L$ and $A^{- (w - 1)} A^{- 1} L = A^{- w} L$ . These are algebraic calculations for the fixed diagram and its two curls, using only [F1] and the definitions. Thus both local curl cases preserve the normalized value.

3.1F1F2F3F4F6F7step 2.1algebra∎

The identity $2$ -braid closes to the two-component unlink, so [F1] gives $P = d^{2}$ and $D = P / d = d$ . Hence [F2] gives $τ_{2} (1) = d^{- 2} d^{2} = 1$ . By [F7], its writhe is $0$ and $c = 2$ , so [F3] returns $- d$ . Then [F6] gives $L (A, Z) = - i^{0} (- 1)^{2} D (- i A, i Z) = - d$ , and therefore $F (A, Z) = - d$ , again matching the output. Here $φ (d) = d$ , and the image of the coefficient relation in [F4] is $i Z (d - 1) = - i (A + A^{- 1})$ , equivalently $Z (d - 1) = - A - A^{- 1}$ . Thus the substitution respects the defining relation; the extension relation $q - q^{- 1} = z$ becomes $q - q^{- 1} = i Z$ as required in $T_{K}$ . All these calculations multiply only by units and never divide by a skein parameter.

Kauffman and Dubrovnik polynomials from BMW algebras — Examples and Counterexamples

2 · Logical flowchart

3 · Definitions, theorems and proofs

4 · Examples, counterexamples and false statements

The leading minus sign in the Dubrovnik conversion is necessary

Statement refuted

Statement references

Diagram

Facts & Assumptions

Refutation

BMW Kauffman and Dubrovnik worked cases

Example

Example references

Facts & Assumptions

Diagram

Proof

Sources