Ordinary Dubrovnik tangle skein algebra

Fix $n\ge 0$ and a rectangle with $n$ distinct ordered endpoints on its top edge and $n$ on its bottom edge. An ordinary $(n,n)$ tangle diagram is a finite generic planar projection in the rectangle of arcs joining these endpoints and finitely many closed curves, with an over/under marking at each transverse double point and no other singularities. Two diagrams are regularly isotopic when related by planar isotopy fixing the boundary and Reidemeister moves II and III. Their regular-isotopy classes form a monoid under vertical stacking, with the $n$ straight strands as its identity. In the fixed crossing chart, $D_{+}$ and $D_{-}$ differ by switching the marked crossing, $D_0$ is the through left/right smoothing, and $D_{\mathrm{\infty }}$ is the cup-cap top/bottom smoothing. The displayed $D_{+}$ has its over-strand running southwest to northeast; with both strands directed downward it is the determinant-positive Artin crossing. A positive curl is a type-I kink with determinant crossing sign $+1$; a negative curl has sign $-1$. Define ${\mathcal{K}\mathcal{T}}_n(T)$ to be the $T$-algebra obtained from the monoid algebra of these classes by imposing, in every interior disk, the Dubrovnik relations $D_{+}-D_{-}=z_D(D_0-D_{\mathrm{\infty }}),\text{positive curl}=a\text{straight strand},\text{negative curl}=a^{-1}\text{straight strand},$ and the free-circle relation that adjoining one disjoint circle with no crossings with the tangle multiplies it by $d$. The symbols $a,z_D,d$ are the images from the coefficient extension. No type-I move is imposed as a regular isotopy; its effect is represented by the displayed curl relations.

Coefficient extension and parameter symbols

Dubrovnik local skein, curl, and circle rules

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

Kauffman, An Invariant of Regular Isotopy