The framed oriented tangle category

Work in the smooth category and fix the slab

$W:=\mathbb{R}\times [0,1]\times [0,1],$

with coordinates $(x,y,t)$ and height function $t$. Its bottom face is $\mathbb{R}\times [0,1]\times \{0\}$, its top face is $\mathbb{R}\times [0,1]\times \{1\}$, and its side faces are $\mathbb{R}\times \{0\}\times [0,1]$ and $\mathbb{R}\times \{1\}\times [0,1]$.

The framed oriented tangle category $\mathcal{T}$ is the strict monoidal category whose objects are the finite sequences $\epsilon =({\epsilon }_1,\dots ,{\epsilon }_m)$ with ${\epsilon }_j\in \{+1,-1\}$, the empty sequence included. For a sequence of length $m$, prescribe boundary points $(j,\frac{1}{2},0)$ on the bottom and $(j,\frac{1}{2},1)$ on the top, $1\le j\le m$, with vertical strand collars and fixed normal vector $(0,1,0)$ on those collars. A morphism $\epsilon \to {\epsilon }^{\mathrm{\prime }}$ is a boundary-relative isotopy class of compact oriented framed $1$-manifolds properly embedded in $W$ and disjoint from the side faces, meeting the bottom face in $m$ points and the top face in $m^{\mathrm{\prime }}$ points at those prescribed positions, equal to the vertical framed collars near both faces, and with the signs ${\epsilon }_1,\dots ,{\epsilon }_m$ at the bottom and ${\epsilon }_1^{\mathrm{\prime }},\dots ,{\epsilon }_{m^{\mathrm{\prime }}}^{\mathrm{\prime }}$ at the top, ordered by the $x$-coordinate. A sign $+1$ means that the oriented tangent points in the increasing $t$ direction; $-1$ means decreasing $t$, at either face. A framing is a homotopy class, relative to the fixed collars, of nonvanishing normal vector fields. Isotopies are ambient isotopies of $W$, constant on the side faces and on the fixed top and bottom collars, that carry the framing class of one tangle to that of the other, fix every boundary point and its framing, and preserve strand orientation. This is the boundary-relative analogue of Oriented links in the three-sphere and ambient isotopy with diagrams in the slab in place of links in the sphere; the oriented crossing signs and kink conventions are those of Oriented Reidemeister moves.

Composition and tensor product. The composition of $\epsilon \to {\epsilon }^{\mathrm{\prime }}$ and ${\epsilon }^{\mathrm{\prime }}\to {\epsilon }^{\mathrm{\prime }\mathrm{\prime }}$ is stacking: put the first tangle in $\mathbb{R}\times [0,1]\times [0,\frac{1}{2}]$, the second in $\mathbb{R}\times [0,1]\times [\frac{1}{2},1]$, and glue along the common boundary. The tensor product is horizontal juxtaposition, placing the first factor to the left of the second and normalizing the ordered endpoints to the prescribed positions for the concatenated sequence. Use horizontal embeddings with disjoint image strips and fixed vertical collars, followed by an order-preserving horizontal adjustment near the boundary. Different such adjustments are isotopic relative to the prescribed framed collars: interpolate the increasing horizontal coordinate maps, whose derivatives remain positive, and extend through the collars. Thus the result is independent of those adjustments. The vertical collars make stacking smooth; different height rescalings and collar lengths are related by increasing height reparametrizations relative to the boundary. These isotopies prove associativity, the identity laws, strict associativity of juxtaposition on isotopy classes, and interchange. The empty sequence is the strict tensor unit. Hence these operations define a strict monoidal category.

Blackboard framing and elementary tangles. When a tangle is drawn in the interior picture plane $y=\frac{1}{2}$ with its bands parallel to that plane, its blackboard framing is represented by the normal to the band surface, equal to $(0,1,0)$ away from small crossing neighborhoods and on the fixed collars. A diagram is a projection to the picture plane; at crossings the bands are separated in the $y$ direction, rather than literally contained in that plane. All elementary tangles below carry the blackboard framing relative to the plane of the picture. The elementary morphisms of $\mathcal{T}$ are:

1. for each pair of signs $(\epsilon ,{\epsilon }^{\mathrm{\prime }})$ the positive crossing $X_{\epsilon ,{\epsilon }^{\mathrm{\prime }}}^{+}\text{\hspace{0.17em}⁣}:(\epsilon ,{\epsilon }^{\mathrm{\prime }})\to ({\epsilon }^{\mathrm{\prime }},\epsilon )$ and the negative crossing $X_{\epsilon ,{\epsilon }^{\mathrm{\prime }}}^{-}\text{\hspace{0.17em}⁣}:(\epsilon ,{\epsilon }^{\mathrm{\prime }})\to ({\epsilon }^{\mathrm{\prime }},\epsilon )$, the two blackboard-framed crossings of the two adjacent strands. The superscript fixes the over/under geometry, independently of their orientations: $X^{+}$ has the geometry of the positive Artin crossing on two $+$ strands, and $X^{-}$ has the inverse geometry. For mixed endpoint signs its oriented crossing sign is reversed. Following the two strands, the crossing sends the left bottom endpoint to the right top endpoint and conversely, which is why the sign sequence is reversed in the target. The two crossings are mutually inverse, $X_{{\epsilon }^{\mathrm{\prime }},\epsilon }^{+}{X}_{\epsilon ,{\epsilon }^{\mathrm{\prime }}}^{-}={\mathrm{id}}_{(\epsilon ,{\epsilon }^{\mathrm{\prime }})}$ and $X_{{\epsilon }^{\mathrm{\prime }},\epsilon }^{-}{X}_{\epsilon ,{\epsilon }^{\mathrm{\prime }}}^{+}={\mathrm{id}}_{(\epsilon ,{\epsilon }^{\mathrm{\prime }})}$;
2. for each sign $\epsilon$ the blackboard-framed cup ${\cup }_{\epsilon }\text{\hspace{0.17em}⁣}:\mathrm{\varnothing }\to (\epsilon ,-\epsilon )$ and cap ${\cap }_{\epsilon }\text{\hspace{0.17em}⁣}:(-\epsilon ,\epsilon )\to \mathrm{\varnothing }$, single arcs with both endpoints on the top face respectively the bottom face, oriented by the sign; these are not invertible;
3. for each sign $\epsilon$ the positive full twist ${\phi }_{\epsilon }^{+}\text{\hspace{0.17em}⁣}:(\epsilon )\to (\epsilon )$ and the negative full twist ${\phi }_{\epsilon }^{-}\text{\hspace{0.17em}⁣}:(\epsilon )\to (\epsilon )$, the blackboard-framed bands carrying one positive respectively negative full twist of the band; they are mutually inverse, ${\phi }_{\epsilon }^{+}{\phi }_{\epsilon }^{-}={\phi }_{\epsilon }^{-}{\phi }_{\epsilon }^{+}=\mathrm{id}$.

All isotopies are ambient isotopies of the slab fixing the boundary points and their framings, so isotopic framed colored tangles represent the same morphism of $\mathcal{T}$.