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BMW relations in the Dubrovnik parameter convention

Statement

Let $S = Z [a^{\pm 1}, z, d^{\pm 1}] / (z (d - 1) - (a - a^{- 1}))$ be the Dubrovnik coefficient ring, and let $T = S [q^{\pm 1}] / (q - q^{- 1} - z) .$ In $T$ put $ρ = a$ , $δ = d$ , and $z_{B} = q - q^{- 1}$ . Then the natural map $S \to T$ is injective and $T$ is a free $S$ -module with basis $1, q$ . Moreover, the BMW coefficient relation $z_{B} (δ - 1) = ρ - ρ^{- 1}$ holds, and the substitution $a = ρ$ , $z_{D} = z_{B}$ , $d = δ$ converts the BMW skein, curl, adjacent untwisting, and loop coefficients to the Dubrovnik relations in the specified chart: $D_{+} - D_{-} = z_{D} (D_{0} - D_{\infty})$ , the curl factors are $a, a^{- 1}$ , and a disjoint trivial circle contributes $d$ . Here $D_{0}$ is through and $D_{\infty}$ is cup-cap. No skein parameter is divided out.

Statement references

BMW presentation and coefficient relation

BMW coefficient extension

Dubrovnik crossing chart and local rules

The fixed smoothing chart

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

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

Poulain d'Andecy, Fusion for the Yang–Baxter Equation and the Braid Group (2020), §2.2

Facts & Assumptions

Given: The BMW presentation over $R_{B}$ , its units $q, ρ, δ$ , and the Dubrovnik chart and parameters $a, z_{D}, d$ .

[F1]

The BMW relation is $g_{i} - g_{i}^{- 1} = z_{B} (1 - e_{i})$ , the coefficient relation is $z_{B} (δ - 1) = ρ - ρ^{- 1}$ , and the local generators satisfy $g_{i} e_{i} = e_{i} g_{i} = ρ^{- 1} e_{i}$ , $e_{i} g_{i \pm 1} e_{i} = ρ e_{i}$ , and $e_{i}^{2} = δ e_{i}$ (BMW algebra in a fixed Kauffman skein convention).

[F2]

In the chosen Dubrovnik chart, $D_{0}$ is the through smoothing, $D_{\infty}$ is the cup-cap smoothing, the crossing rule is $D_{+} - D_{-} = z_{D} (D_{0} - D_{\infty})$ , and the two curl factors are $a, a^{- 1}$ (Kauffman regular-isotopy and Dubrovnik conventions).

[F3]

The bracket chart labels the left/right pairing as the A-smoothing and the top/bottom pairing as the B-smoothing for an over-strand directed southwest to northeast (Kauffman bracket smoothing and loop value).

[F4]

The coefficient extension is $T = S [q^{\pm 1}] / (q - q^{- 1} - z)$ , with $z_{B} = q - q^{- 1}$ , $z_{D} = z_{B}$ , $ρ = a$ , and $δ = d$ (BMW coefficient extension for Dubrovnik parameters).

Proof

1.1F4givenconstructalgebra

The relation defining $T$ is $q - q^{- 1} = z$ , hence $q^{2} - z q - 1 = 0$ and $q^{- 1} = q - z$ . Therefore $T ≅ S [q] / (q^{2} - z q - 1)$ . Division by this monic quadratic gives a unique remainder $u + v q$ with $u, v \in S$ , so $T$ is free over $S$ on $1, q$ and the constant inclusion is injective.

2.1F1F2F4step 1.1algebra

With $ρ = a$ , $δ = d$ and $z_{B} = z_{D} = z$ , the BMW coefficient relation becomes exactly $z (d - 1) = a - a^{- 1}$ , the defining relation of $S$ . Thus every coefficient assignment is compatible. No skein variable is inverted.

3.1F1F2F3F4step 2.1algebra

In the fixed chart the downward positive Artin crossing has over-strand from top right to bottom left. Its determinant sign is positive, and its orientation-preserving smoothing is the through pairing $D_{0}$ ; the other smoothing is the cup-cap $D_{\infty}$ . Consequently the BMW relation maps to $G_{i} - G_{i}^{- 1} = z_{B} (1 - E_{i}) = z_{D} (D_{0} - D_{\infty})$ , as required. The bracket's names $A$ and $B$ remain those of [F3]; in this chart the Dubrovnik names $D_{0}, D_{\infty}$ are respectively $A, B$ .

4.1F1F2step 3.1algebra∎

Composing the positive crossing with a cup-cap reverses the direction of exactly one incident branch when the resulting self-crossing is oriented along its strand. The determinant sign is therefore negative and its curl coefficient is $a^{- 1} = ρ^{- 1}$ , agreeing with $g_{i} e_{i} = e_{i} g_{i} = ρ^{- 1} e_{i}$ . The inverse crossing gives $a = ρ$ . In $E_{i} G_{i \pm 1} E_{i}$ the surviving self-curl has the opposite sign and coefficient $a = ρ$ , agreeing with the adjacent untwisting relation. A double cup-cap creates one trivial circle, so its coefficient is $d = δ$ . These checks distinguish a curl after bending from a positive braid closure, whose strands retain their downward orientations and whose curl coefficient is $a$ .

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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, Definition 2.6 and Definition 3.1 (standard reference, not scraped)
Loïc Poulain d'Andecy, Fusion for the Yang–Baxter Equation and the Braid Group, §2.2 (standard reference, not scraped)