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Kauffman regular-isotopy and Dubrovnik conventions

Definition

Let $R_{D} = Z [a^{\pm 1}, z, d^{\pm 1}] / (z (d - 1) - (a - a^{- 1})),$ where $z$ is not inverted. A nonempty unoriented regular link diagram is a finite generic planar projection of one or more circles with only transverse double crossings and an over/under marking at each crossing. Use the crossing chart in which the over-strand runs southwest to northeast. Its $D_{0}$ smoothing joins the left endpoints and the right endpoints (the through pairing), and its $D_{\infty}$ smoothing joins the two top endpoints and the two bottom endpoints (the cup-cap pairing). Thus $D_{0}$ is the orientation-preserving smoothing when both crossing strands point downward and the displayed crossing has positive determinant sign. These are respectively the $A$ - and $B$ -smoothings in the stated bracket chart. A positive curl is a type-I kink whose oriented crossing has determinant sign $+ 1$ ; a negative curl has determinant sign $- 1$ . The sign is unchanged if the orientation of the whole component is reversed.

The Dubrovnik regular-isotopy evaluation $P$ is normalized by $P (\emptyset) = 1$ , $P (K ⊔ ◯) = d P (K)$ , positive and negative curl factors $a$ and $a^{- 1}$ , and $P (D_{+}) - P (D_{-}) = z (P (D_{0}) - P (D_{\infty})) .$ Here $D_{+}$ and $D_{-}$ differ only by switching the displayed crossing. For a nonempty oriented link diagram $K$ , let $c (K)$ be its number of components and let $w (K)$ be the sum of the determinant crossing signs. Define $D_{K} (a, z) = d^{- 1} P (K),$ $L_{K} (A, Z) = - i^{w (K)} (- 1)^{c (K)} D_{K} (- i A, i Z), i^{2} = - 1,$ and $F_{K} (A, Z) = A^{- w (K)} L_{K} (A, Z) .$ Thus $D$ has unknot value $1$ , and the universal relation defining $R_{D}$ is retained even when $z$ is not invertible. These are conventions and normalizations; ambient-isotopy invariance of $F$ is a separate theorem.

Definition references

Determinant crossing sign and writhe

Kauffman bracket smoothing chart

Scalar regular-isotopy evaluation and universal coefficient relation

Kauffman, An Invariant of Regular Isotopy (1990)

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)

Depends on

Used by

Compute the Kauffman polynomial from a BMW braid closure Algorithm
The leading minus sign in the Dubrovnik conversion is necessary Counterexample
BMW coefficient extension for Dubrovnik parameters Definition
Ordinary Dubrovnik tangle skein algebra Definition
BMW Kauffman and Dubrovnik worked cases Example
BMW relations in the Dubrovnik parameter convention Lemma
BMW tangle skein spanning by descending matchings Lemma
Kauffman and Dubrovnik writhe-component conversion Lemma
The BMW presentation maps to the Dubrovnik tangle algebra Lemma
The BMW Markov trace gives the Dubrovnik link invariant Theorem
The Kauffman and Dubrovnik polynomials agree under the stated conversion Theorem

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Sources

Louis H. Kauffman, An Invariant of Regular Isotopy, Transactions of the AMS 318(2) (1990), §§II, VI, VII, printed pp. 420–429 and 452–469 (standard reference, not scraped)
Frederick M. Goodman and Holly Hauschild, Affine Birman–Wenzl–Murakami Algebras and Tangles in the Solid Torus, §§2.1–2.3 and Definition 3.1 (standard reference, not scraped)
Loïc Poulain d’Andecy, Fusion for the Yang–Baxter Equation and the Braid Group, Winter Braids Lecture Notes 7 (2020), §2.2 (standard reference, not scraped)