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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.

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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)