ScalingStacks

We introduce two 2-categories $\mathbb{K}$ and $\widehat{\mathbb{K}}.$

Objects of $\mathbb{K}$ are non-negative integers, 1-morphisms from $n$ to $m$ are objects of $\mathcal{K}_{n}^{m},$ and 2-morphisms between $M,N\in\mathcal{K}_{n}^{m}$ are $\mathrm{Hom}_{\mathcal{K}_{n}^{m}}(M,N),$ grading-preserving morphisms of complexes of bimodules up to chain homotopies. Composition of 1-morphisms is given by tensor product of complexes.

The 2-category $\widehat{\mathbb{K}}$ has the same objects and 1-morphisms as $\mathbb{K}$ but the 2-morphisms are

that is, the morphisms are all homomorphisms (not just grading-preserving), and each homomorphism is identified with its negative. The set of 2-morphisms between two 1-morphisms is no longer an abelian group.

We next construct a 2-functor ${\mathcal{F}}:{\mathcal{C}}\to\widehat{\mathbb{K}}.$ This 2-functor takes object $s$ of ${\mathcal{C}}$ to the object $|s|$ of $\widehat{\mathbb{K}}.$ It takes generating 1-morphisms of ${\mathcal{C}}$ to complexes of bimodules in the same way as in [9, Sections 2.7, 3.4]. Recall that a U-turn $b$ (and, more generally, any flat tangle) is taken to the complex $0\longrightarrow{\mathcal{F}}(b)\longrightarrow 0$ where ${\mathcal{F}}(b)$ is the bimodule associated to $b.$ A crossing $r$ gives rise to its two resolutions $r(0)$ and $r(1)$ and a grading-preserving bimodule map $\psi:{\mathcal{F}}(r(0))\longrightarrow{\mathcal{F}}(r(1))\{-1\}.$ Then ${\mathcal{F}}(r)$ is defined as the complex

with a suitable grading shift computed from the orientation of $r$ near its crossing.

The 2-functor ${\mathcal{F}}$ takes composition of 1-morphisms to the tensor product of complexes:

where $a,$ respectively $b,$ has $2n$ bottom, respectively, $2n$ top endpoints.

To a Reidemeister move between diagrams $a$ and $b$ (see figure 1) we assign an isomorphism of bimodule complexes ${\mathcal{F}}(a)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathcal{F}}(b)$ constructed in [9, Section 4]. The Reidemeister III move has several versions, depending on the directions of overcrossings. In [9] we described an isomorphism in $\mathcal{K}_{n}^{n}$ between complexes ${\mathcal{F}}(a)$ and ${\mathcal{F}}(b)$ for only one version of this move. Other versions can be expressed via compositions of this version with isotopies and type II moves. The compositions induce isomorphisms between ${\mathcal{F}}(a)$ and ${\mathcal{F}}(b)$ which we assign to these other version of the Reidemeister III move. Note that, since Reidemeister move diagrams in figure 1 are either braids or composition of a U-turn and a braid, a grading-preserving isomorphism between ${\mathcal{F}}(a)$ and ${\mathcal{F}}(b)$ is unique up to minus sign, by Corollaries 2, 4.

The diagrams of birth, death, saddle point, and T-move do not involve crossings. These movies can be viewed as presentations of surfaces embedded in $\mathbb{R}^{3},$ and to them we assign bimodule homomorphisms using the construction of Proposition 5 of [9]. To the birth 2-morphism we assign the unit map $\iota:\mathbb{Z}\to{\mathcal{A}},$ to the death 2-morphism the counit map $\epsilon:{\mathcal{A}}\to\mathbb{Z}.$ More precisely, birth and death moves happen inside $\mathrm{Vert}_{n}$ diagrams, and the maps are

To the diagrams of saddle point cobordisms between $\mathrm{Vert}_{n}$ and $\cup_{i,n-1}\cap_{i,n}$ we assign bimodule maps (2), (3). These maps are, up to sign, the only maps of degree $1$ between ${\mathcal{F}}(\mathrm{Vert}_{n})\cong H^{n}$ and ${\mathcal{F}}(\cup_{i,n-1}\cap_{i,n})$ that generate the abelian group (isomorphic to $\mathbb{Z}$) of all degree $1$ homomorphisms between these bimodules.

The natural isotopy between the two diagrams $a,b$ in the H-move (figure 3) induces an isomorphisms of complexes ${\mathcal{F}}(a)\cong{\mathcal{F}}(b),$ which we assign to this 2-morphism.

A frame of an N-move between diagrams $a$ and $b$ contains two little squares, and each square is either a U-turn or a crossing. The N-move is an isotopy from $a$ to $b.$ This isotopy induces a canonical isomorphism of complexes ${\mathcal{F}}(a)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathcal{F}}(b)$ (see [9, Sections 4.1, 4.2]), which we assign to the N-move.

This theorem is proved in Section 5.

$\mathcal{C}$ is a combinatorial realization of the 2-category $\mathcal{T}$ of tangle cobordisms, that is, the natural 2-functor $\mathcal{C}\to\mathcal{T}$ is an equivalence of 2-categories, see [1]. This result is also valid for oriented tangles.

As a corollary, we obtain a 2-functor, also denoted ${\mathcal{F}},$ from the 2-category $\mathcal{T}$ of even unframed oriented tangle cobordisms to $\widehat{\mathbb{K}}.$ The homomorphism of complexes of graded $(H^{m},H^{n})$-bimodules assigned to the cobordism $S$ between $(m,n)$-tangles has degree $n+m-\chi(S),$ where $\chi(S)$ is the Euler characteristic of $S.$