ScalingStacks

The category $\Rep(U_{q}(\mathfrak{gl}_{N}))$ of finite-dimensional $U_{q}(\mathfrak{gl}_{N})$-modules is a ribbon category and thus provides Reshetikhin–Turaev invariants of framed oriented tangles with components labeled by objects of $\Rep(U_{q}(\mathfrak{gl}_{N}))$. A framed oriented link $L$ labeled by the $U_{q}(\mathfrak{gl}_{N})$-module $V={\mathbb{C}}^{N}(q)$ yields an endomorphism of ${\mathbb{C}}(q)$, the tensor unit in $\Rep(U_{q}(\mathfrak{gl}_{N}))$, which is just multiplication by the $\mathfrak{gl}_{N}$ link polynomial of $L$.

While we will focus on invariants of links labeled by $V$, it is convenient to also consider the fundamental modules $\bigwedge^{k}(V)$ and their duals. Together, these generate the full monoidal subcategory $\mathrm{Fund}(U_{q}(\mathfrak{gl}_{N}))$, which admits a graphical presentation and which recovers $\Rep(U_{q}(\mathfrak{gl}_{N}))$ upon idempotent completion.

The ${\mathbb{C}}(q)$-linear pivotal category $\boldsymbol{\mathrm{Web}}_{N}$ has objects given by finite sets of points in an interval ${[0,1]}$, each labeled by an element of $\{n,n^{*}|n\in{\mathbb{Z}}_{>0}\}$. The morphisms are ${\mathbb{C}}(q)$-linear combinations of webs: oriented trivalent graphs, properly embedded in ${[0,1]}^{2}$, with edges labeled by a non-negative integer flow, considered up to isotopy relative to the boundary and certain local relations, including those shown in (2.1). The source and target of a web are determined by its intersections with ${[0,1]}\times\{0\}$ and ${[0,1]}\times\{1\}$, with downward oriented boundary points of label $n$ being recorded as $n^{*}$. Composition is given by the bilinear extension of stacking webs and the tensor product is given on objects by concatenating labeled intervals and on morphisms by the bilinear extension of placing webs side by side.

The morphisms in $\boldsymbol{\mathrm{Web}}_{N}$ are generated under composition, tensor product, and duality by identity morphisms and trivalent merge and split vertices:

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The merge and split vertices encode the natural $U_{q}(\mathfrak{gl}_{N})$-intertwiners

$$\textstyle\bigwedge^{a}(V)\otimes\bigwedge^{b}(V)\to\bigwedge^{a+b}(V)\quad\text{and}\quad\bigwedge^{a+b}(V)\to\bigwedge^{a}(V)\otimes\bigwedge^{b}(V),$$

respectively. The local relations in $\boldsymbol{\mathrm{Web}}_{N}$ include

(2.1)

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together with the reflections of these relations in a vertical line. Edges labeled zero are to be erased and edges labeled by negative integers or integers greater than $N$ force the morphism to be the zero morphism. The local relations ensure that $\boldsymbol{\mathrm{Web}}_{N}\cong\mathrm{Fund}(U_{q}(\mathfrak{gl}_{N}))$ as ${\mathbb{C}}(q)$-linear pivotal categories, see [CKM14, QS19, TVW17]. In the following, we also consider an integral version of $\boldsymbol{\mathrm{Web}}_{N}$, which is defined over ${\mathbb{Z}}[q^{\pm 1}]$ and also satisfies the relations (2.1).

Foams provide a framework for a combinatorial description of Khovanov–Rozansky link homologies, in a similar way as webs are useful for the type A Reshetikhin-Turaev invariants. We will use $\mathfrak{gl}_{N}$-foams constructed via the combinatorial evaluation formula for closed foams due to Robert–Wagner [RW20]. More precisely, we will organize these $\mathfrak{gl}_{N}$-foams into a monoidal bicategory $\boldsymbol{\mathrm{Foam}}_{N}$ which categorifies the integral form of $\boldsymbol{\mathrm{Web}}_{N}$.

The graded, additive monoidal bicategory $\boldsymbol{\mathrm{Foam}}_{N}$ has objects given by finite sets of points in ${[0,1]}$, each labeled by an element of $\{n,n^{*}|n\in{\mathbb{Z}}_{>0}\}$. The 1-morphisms are (formal direct sums of formal grading shifts of) webs, properly embedded in ${[0,1]}^{2}$ and connecting boundary points of appropriate labels. Note that webs are not considered up to any relations in $\boldsymbol{\mathrm{Foam}}_{N}$. The 2-morphisms are (matrices of degree zero) ${\mathbb{Z}}$-linear combinations of $\mathfrak{gl}_{N}$-foams in ${[0,1]}^{3}$, considered up to isotopy relative to the boundary and certain local relations, as defined in [ETW18, Section 2]. The three compositions are given by (the bilinear extension of) stacking these topological objects along the three interval directions.

Foams are the natural notion of cobordisms between webs and the relations between $2$-morphisms in $\boldsymbol{\mathrm{Foam}}_{N}$ are chosen so that the defining web equalities in $\boldsymbol{\mathrm{Web}}_{N}$ can be lifted to explicit web isomorphisms in $\boldsymbol{\mathrm{Foam}}_{N}$. We refer to [ETW18] for a rigorous definition of $\mathfrak{gl}_{N}$-foams, as well as a complete description of the relations between them, and a survey of various flavors of $\boldsymbol{\mathrm{Foam}}_{N}$. Here we only comment on aspects relevant to the rest of this paper.

Foams are represented by 2-dimensional cell complexes, such that every point has a neighborhood either modelled on ${\mathbb{R}}^{2}$, three half-planes meeting in a line, or the cone on the 1-skeleton of a tetrahedron. Such cone points are called singular vertices of the foam. The points on the line in the second case form a seam of the foam, and the connected components of the set of manifold points are called the facets of the foam. An example of a foam with six singular vertices is shown in Figure 1. The facets are oriented and labeled by positive integers. If three facets meet along a seam, then two of their labels, say $a$ and $b$, sum to the third, $a+b$. The orientation of the seam agrees with the orientation induced by the $a$ and $b$ facets, and disagrees with the $a+b$ facet.

Each facet of a foam in $\boldsymbol{\mathrm{Foam}}_{N}$ admits an action of the algebra of symmetric functions $\Lambda$. This is to say that facets may be decorated by points labeled by symmetric functions, which are allowed to move freely on facets. A point labeled by a product $fg\in\Lambda$ may be split into two points labeled $f$ and $g$ respectively, and a foam with a point labeled $f+g\in\Lambda$ may be split into a sum of foams with points labeled $f$ and $g$ respectively. The $\Lambda$-actions on adjacent facets are compatible in the sense that $f\in\Lambda$ on an $a+b$ facet may be moved across a seam, where it distributes into $\Delta(f)\in\Lambda\otimes\Lambda$ acting on the adjacent $a$ and $b$ facets. The degree of a foam is computed as twice the degree of the symmetric function decoration, minus a weighted Euler characteristic, depending on facet labels.

$\boldsymbol{\mathrm{Foam}}_{N}$ is designed to have finite-dimensional spaces of $2$-morphisms, and in particular, the $\Lambda$-action on each $a$-facet factors through a finite-dimensional quotient, namely $H^{*}(\mathrm{Gr}({\mathbb{C}}^{a}\subset{\mathbb{C}}^{N}))$, the cohomology ring of the Grassmannian of $a$-dimensional subspaces of ${\mathbb{C}}^{N}$, which is obtained as quotient of $\Lambda$ by the ideal $\langle h_{N-a+i}|i>0\rangle$ generated by sufficiently large complete symmetric functions. In the case of a $1$-labeled facet, the symmetric function $e_{1}=h_{1}$ is called the dot.

The construction of Khovanov–Rozansky link homologies now proceeds in two steps. The first step is a functor that sends link diagrams to chain complexes in $\boldsymbol{\mathrm{Foam}}_{N}$ and link cobordisms to chain maps, which depend only on the isotopy type of the cobordism up to homotopy. The second step evaluates such a chain complex to a bigraded abelian group through a representable functor and taking homology.

In one direction, by forgetting the condition on the projection and ignoring the crossing order, this category is equivalent to the usual category of all embedded, framed oriented links and link cobordisms. In the other direction, the category ${\mathbb{R}}^{3}\boldsymbol{\mathrm{Link}}^{\circ}$ is equivalent to the category whose objects are link diagrams and whose morphisms are sequences of Reidemeister moves, Morse moves, planar isotopies, and formal reorderings, considered up to Carter–Rieger–Saito movie moves [CS93, CRS97].

We will now describe the construction of a functor $\left\llbracket-\right\rrbracket\colon{\mathbb{R}}^{3}\boldsymbol{\mathrm{Link}}^{\circ}\to\boldsymbol{\mathrm{K}}^{b}(\boldsymbol{\mathrm{Foam}}_{N})$, with target given by the bounded homotopy category of $\boldsymbol{\mathrm{Foam}}_{N}$. In particular, the functor sends link diagrams to certain bounded chain complexes of webs and foams. On single, $1$-labeled crossings, it is defined as:

(2.3)

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The underlined term is placed in homological degree zero. We call the non-identity webs that appear here thick edges. The differentials in both complexes are given by the combinatorially simplest foam between the two shown webs. We call them unzip and zip foams respectively. The reader be warned that the assignments in (2.3) differ from the conventions in [KR08, Fig. 46] by a $q$-grading shift of magnitude $(N-1)w$, where $w$ denotes the writhe of the diagram; the latter further differ from the conventions in [ETW18, Equation (3.2)] by mirroring.

A link diagram with several crossings (in a specified order) is sent to the chain complex constructed from the formal tensor product of the crossing complexes (2.3) (in that order) by gluing its resolutions into the link diagram in place of the original crossings.

The chain complexes associated to link diagrams which differ only by Reidemeister moves are homotopy equivalent, see Sections 3.3–3.5. Similarly, one can define chain maps for Morse moves. However, a highly non-trivial fact is that there exists a coherent choice for such chain maps.

In fact, Theorem 2.4 holds in much greater generality, including colored links and the equivariant framework mentioned in Remark 2.2. More importantly for us, the theorem holds locally, i.e. for tangle diagrams and tangle cobordisms.