ScalingStacks

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:

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.