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The BMW presentation maps to the Dubrovnik tangle algebra

Statement

Let $T$ be the coefficient extension and parameter substitution in the BMW-to-Dubrovnik dictionary, and let ${K T}_{n} (T)$ be the ordinary Dubrovnik tangle skein algebra: its diagrams are finite generic projections of $n$ arcs with fixed ordered top and bottom endpoints and closed curves, modulo planar isotopy, Reidemeister II and III, the Dubrovnik skein relation, the two curl relations, and the free-circle relation with parameters $z_{D}, a, a^{- 1}, d$ . In its fixed chart, $D_{0}$ is through left/right and $D_{\infty}$ is cup-cap top/bottom. For every $n \geq 2$ , send the BMW generator $g_{i}$ to the positive Artin crossing between strands $i, i + 1$ , its inverse to the negative crossing, and $e_{i}$ to the adjacent cup-cap tangle. This assignment extends to a unital $T$ -algebra homomorphism $Φ_{n} : B M W_{n} (T; q, ρ, δ) \to {K T}_{n} (T)$ . For $n = 0, 1$ , define the unital $T$ -algebra homomorphism by sending each scalar to that scalar times the empty or straight tangle, respectively. This claim asserts well-definedness only; it does not assert surjectivity or injectivity.

Statement references

BMW algebra presentation

Dubrovnik tangle rules

Coefficient and parameter substitution

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

Kauffman, An Invariant of Regular Isotopy

Facts & Assumptions

Given: The BMW presentation over $T$ , the Dubrovnik coefficient extension, and the rectangular regular-isotopy tangle algebra described in the statement.

[F1]

The BMW generators satisfy the braid and distant-commutation relations, the distant cup-cap relations, $e_{i} e_{i \pm 1} e_{i} = e_{i}$ , $g_{i} g_{i \pm 1} e_{i} = e_{i \pm 1} e_{i}$ , $e_{i} g_{i \pm 1} g_{i} = e_{i} e_{i \pm 1}$ , $g_{i} - g_{i}^{- 1} = z_{B} (1 - e_{i})$ , $g_{i} e_{i} = e_{i} g_{i} = ρ^{- 1} e_{i}$ , $e_{i} g_{i \pm 1} e_{i} = ρ e_{i}$ , and $e_{i}^{2} = δ e_{i}$ (BMW algebra in a fixed Kauffman skein convention).

[F2]

Under $a = ρ$ , $z_{D} = z_{B}$ , and $d = δ$ , the scalar relation holds, $D_{0}$ is the through smoothing, $D_{\infty}$ is the cup-cap smoothing, the crossing rule is $D_{+} - D_{-} = z_{D} (D_{0} - D_{\infty})$ , the curl factors are $a, a^{- 1}$ , and a free circle contributes $d$ (BMW relations in the Dubrovnik parameter convention, Kauffman regular-isotopy and Dubrovnik conventions).

[F3]

The fixed crossing chart identifies the left-right pairing as the A-smoothing and the top-bottom pairing as the B-smoothing (Kauffman bracket smoothing and loop value).

[F4]

In the generator schematic, top endpoints are above bottom endpoints, the separated strand at a crossing is the over-strand, and stacking is read from top to bottom; the positive crossing is calibrated by the determinant-sign definition (Oriented crossing signs, diagram writhe, and blackboard framing).

Diagram

The positive crossing and the cup-cap generator are shown with stacking read from top to bottom. The white gap marks the under-strand.

Proof

1.1F1given

The inverse of the positive crossing is its negative crossing: stacking them creates a type-II bigon, which cancels by regular isotopy. For $∣ i - j ∣ \geq 2$ , the supports of $g_{i}$ and $g_{j}$ are disjoint, so they commute; the supports of $g_{i}$ and $e_{j}$ are disjoint, so those commute as well; and two distant cup-cap tangles commute by planar isotopy. The adjacent braid relation is the type-III move on three monotone strands, and the two possible orders of stacking the cup-cap tangles in $e_{i} e_{i \pm 1} e_{i} = e_{i}$ reduce by planar isotopy to the same single cup and cap. These checks preserve the ordered boundary endpoints.

1.2F1F2F3F4algebra

The BMW skein relation maps to $G_{i} - G_{i}^{- 1} = z_{B} (1 - E_{i}) = z_{D} (D_{0} - D_{\infty})$ : here $z_{B} = z_{D}$ , the identity is the through smoothing $D_{0}$ , and $E_{i}$ is the cup-cap smoothing $D_{\infty}$ . In the fixed southwest-to-northeast over-strand chart, write the projected oriented tangent coordinates as $(t, x)$ with $t$ increasing downward. The over and under tangents of the downward crossing can be taken as $(1, - 1)$ and $(1, 1)$ ; their determinant is $2 > 0$ by [F4], and its oriented smoothing is through. The bracket names in [F3] are $A, B$ for these two pairings, respectively.

2.1F1F2F4step 1.2algebra

When the positive crossing is followed by a cup-cap, the two branches belong to the same bent strand. Relative to their downward braid orientations exactly one branch reverses, so its determinant self-crossing sign is negative. Thus $G_{i} E_{i} = E_{i} G_{i} = a^{- 1} E_{i} = ρ^{- 1} E_{i}$ ; the negative crossing gives the positive self-curl and coefficient $a = ρ$ . For $E_{i} G_{i + 1} E_{i}$ , follow the surviving strand from the top position $i + 2$ : it traverses the crossing from position $i + 2$ to $i + 1$ , turns along the lower cup to position $i$ , returns up the uncrossed strand at position $i$ to the upper cap, and traverses the other crossing branch from $i + 1$ to $i + 2$ . Both encounters are downward in the crossing disk, so the self-crossing has positive determinant sign and gives $a E_{i} = ρ E_{i}$ . For $E_{i} G_{i - 1} E_{i}$ , start at the top position $i - 1$ : go down the crossing's left branch, around the lower cup and up position $i + 1$ , then around the upper cap and down the crossing's right branch. Again both crossing branches are traversed downward, giving the same positive curl. Stacking $E_{i}$ twice creates a disjoint trivial circle and leaves $E_{i}$ , so $E_{i}^{2} = d E_{i} = δ E_{i}$ . The self-curl sign after a cup is distinct from the positive self-curl formed by closing a positive two-strand braid.

3.1F1givenstep 1.1step 1.2step 2.1

For the remaining adjacent relations, use three consecutive positions $1, 2, 3$ . In $G_{1} G_{2} E_{1}$ , the upper cup joins top endpoints $2, 3$ , the lower cap joins bottom endpoints $1, 2$ , and the remaining strand joins top $1$ to bottom $3$ . That strand passes under both arms of the upper cup. Slide the cup toward the top: its two crossings with that strand form a type-II pair and cancel, leaving the planar tangle $E_{2} E_{1}$ . Hence $G_{1} G_{2} E_{1} = E_{2} E_{1}$ . To reverse stacking, reflect top to bottom AND interchange over and under at every crossing. This is a half-turn in three-space about a horizontal axis in the projection plane; it preserves regular isotopy, fixes $G_{j}$ and $E_{j}$ , and reverses products. It therefore gives $E_{1} G_{2} G_{1} = E_{1} E_{2}$ . Likewise, left-to-right reflection together with over/under interchange fixes crossing chirality and sends $G_{1}, G_{2}$ to $G_{2}, G_{1}$ and $E_{1}, E_{2}$ to $E_{2}, E_{1}$ . Applying it to the two identities gives $G_{2} G_{1} E_{2} = E_{1} E_{2}$ and $E_{2} G_{1} G_{2} = E_{2} E_{1}$ . Translating the three-position rectangle gives both signs of the adjacent index in [F1]. Neither operation is a lone planar reflection, which would invert the crossings. Together with steps 1.1, 1.2 and 2.1, these verify every defining relation for $n \geq 2$ .

4.1F1givenalgebra∎

For $n = 0$ and $n = 1$ , the BMW algebra is the scalar algebra $T$ by [F1], with no crossing or cup-cap generators. Sending a scalar to the same scalar times the empty tangle or the straight tangle is unital and $T$ -linear, so it defines the stated boundary homomorphism.

Depends on

Used by

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Frederick M. Goodman and Holly Hauschild, Affine Birman–Wenzl–Murakami Algebras and Tangles in the Solid Torus, §§2.2–2.3 and Definition 3.1 (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)