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BMW tangle skein spanning by descending matchings

Statement

For each integer $n \geq 0$ , let $S = Z [a^{\pm 1}, z, d^{\pm 1}] / (z (d - 1) - (a - a^{- 1}))$ and $T = S [q^{\pm 1}] / (q - q^{- 1} - z)$ , with $z_{D} = q - q^{- 1}$ . Let ${K T}_{n} (T)$ be the ordinary Dubrovnik tangle skein algebra formed from regular-isotopy classes with crossing rule $D_{+} - D_{-} = z_{D} (D_{0} - D_{\infty})$ , curl factors $a, a^{- 1}$ , and free-circle factor $d$ , where $D_{0}$ is through and $D_{\infty}$ is cup-cap. Number its $2 n$ boundary endpoints in cyclic order as $1, 2, \dots, n, \overset{ˉ}{n}, \dots, \overset{ˉ}{2}, \overset{ˉ}{1}$ . A connector is a perfect matching of these endpoints. Orient each open strand from its lower-numbered endpoint to its higher-numbered endpoint and order the open strands by their initial endpoints. Place the finitely many closed strands after the open strands in any fixed order, with a chosen orientation and initial point for each. A diagram is totally descending if, in this traversal order, the first branch encountered at each crossing is the overpassing branch. Then ${K T}_{n} (T)$ is spanned by one loop-free, self-crossing-free totally descending tangle for each connector. These representatives can have crossings between distinct strands; their crossing data are fixed by the total traversal order. For $n = 0$ the single representative is the empty tangle, and for $n = 1$ it is the straight strand.

Statement references

The Dubrovnik tangle skein algebra

Dubrovnik skein, curl, and circle rules

Goodman and Hauschild, Affine Birman–Wenzl–Murakami Algebras and Tangles in the Solid Torus, §5.1

Kauffman, An Invariant of Regular Isotopy

Facts & Assumptions

Given: The finite rectangular tangle diagrams, their fixed endpoint order, the Dubrovnik skein and curl relations, and the generator interpretation from the preceding map.

[F1]

A tangle diagram is a generic projection of finitely many arcs with fixed boundary endpoints and closed curves; regular isotopy is generated by planar isotopy and Reidemeister II and III (Ordinary Dubrovnik tangle skein algebra).

[F2]

The tangle coefficient rules are $D_{+} - D_{-} = z_{D} (D_{0} - D_{\infty})$ , positive and negative curls multiply by $a, a^{- 1}$ , and a disjoint trivial circle multiplies by $d$ (Kauffman regular-isotopy and Dubrovnik conventions).

Proof

1.1F1F2base

If a tangle diagram has no crossings, every closed component is a disjoint trivial circle and the remaining arcs form a crossingless connector; removing each circle by [F2] leaves a scalar multiple of its standard matching tangle. This is the base case for induction on the lexicographically ordered pair (crossing count, bad-crossing count).

1.2F1F2ihinduction

For a diagram with crossings, choose a finite traversal order of its components and orient each arc from its smaller boundary label to its larger one; for closed components use any finite encoded orientation and basepoint. A crossing is bad when its first encountered branch is under. If there is a bad crossing, the skein relation expresses the diagram as its switched diagram plus two smoothing terms. Switching preserves the crossing count and lowers the bad-crossing count by one; each smoothing lowers the crossing count by one. Thus every term is lower in the stated lexicographic order, and the induction hypothesis expresses each as a linear combination of standard descending representatives. The choices here are finite choices made from the encoded diagram, not an appeal to an infinite choice principle.

1.3F1F2algebra

If there is no bad crossing, the diagram is totally descending. Earlier components pass over later ones, so separate their interiors into successive height layers, keeping the boundary endpoints fixed. Within one layer, the first-encounter-over condition lets the already traversed part of a component be pulled above its untraversed part: it is an unknotted arc relative to its endpoints, or an unknotted closed curve. Straighten each arc to a simple planar arc in its layer and each closed curve to a circle by ambient isotopy. This simplification may use type-I moves on open arcs as well as on closed curves; it is not asserted to be regular isotopy. In a finite generic diagram movie, planar isotopy and moves II and III preserve the skein class, while each deletion of a positive or negative curl contributes $a$ or $a^{- 1}$ by [F2] (an insertion contributes the inverse factor). Thus the original diagram equals $a^{r}$ times the resulting layered diagram for some integer $r$ . Separate and delete its $c$ trivial closed circles by the free-circle rule, leaving $a^{r} d^{c}$ times a loop-free, self-crossing-free descending diagram with the same open connector.

2.1F1F2step 1.1step 1.2step 1.3discharge-induction: step 1.2∎

The base case, induction step, and descending terminal case together show by well-founded induction that every diagram reduces to a linear combination of standard representatives. For each connector choose simple arcs in the rectangle and put their interiors in height layers ordered by their initial endpoints; a generic projection gives a loop-free, self-crossing-free descending representative. Any two such representatives with that connector can be put in the same ordered layers. In a disk, two simple arcs with the same endpoints are isotopic relative to those endpoints; apply this within each layer, leaving the other layers fixed and retaining the boundary collars. These isotopies keep each arc simple and its blackboard framing untwisted, so their generic projections use only planar isotopy and moves II and III, with no curl factor. Thus the loop-free, self-crossing-free representatives are unique up to regular isotopy. There are $(2 n - 1)!!$ perfect matchings on $2 n$ endpoints, obtained by pairing the first endpoint in $2 n - 1$ ways and repeating; for $n = 0$ the empty matching is unique and for $n = 1$ the straight strand is unique.

Depends on

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Sources

Frederick M. Goodman and Holly Hauschild, Affine Birman–Wenzl–Murakami Algebras and Tangles in the Solid Torus, §§2.2–2.3 and 5.1 (standard reference, not scraped)
Louis H. Kauffman, An Invariant of Regular Isotopy, Transactions of the AMS 318(2) (1990), §§VI–VII (standard reference, not scraped)