ScalingStacks

In the following we give an outline of the construction. We start with the Khovanov–Rozansky link homologies, which are categorifications of the $\mathfrak{gl}_{N}$ quantum link invariants of Reshetikhin–Turaev [RT90]. These link homologies take the shape of functors

which were constructed by Ehrig–Tubbenhauer–Wedrich in [ETW18] following earlier work on functoriality by Bar-Natan [BN05], Clark–Morrison–Walker [CMW09], and Blanchet [Bla10], and using technology developed by Robert–Wagner [RW20] and Rose–Wedrich [RW16] following Mackaay–Stošić–Vaz [MSV09], Lauda–Queffelec–Rose [LQR15], and Queffelec–Rose [QR16].

It is worth emphasizing that the functors $\mathrm{KhR}_{N}$ considered here are defined combinatorially and normalized to be sensitive to framing changes, in contrast to earlier incarnations of Khovanov–Rozansky homology. In the following, all links are oriented and framed and all link cobordisms are oriented.

The first step in our construction is to show that Khovanov–Rozansky homologies make sense as functorial invariants of links in $S^{3}$, rather than just in ${\mathbb{R}}^{3}$. From the point of view of link embeddings and link cobordisms, there is not much difference between these two cases. A generic link embedding will miss the point $\infty$ if we consider ${\mathbb{R}}^{3}=S^{3}\setminus\{\infty\}$ and a generic link cobordism embedded in $S^{3}\times{[0,1]}$ will miss $\{\infty\}\times{[0,1]}$. However, the analogous statement is no longer true for isotopies of link cobordisms. While link embeddings and their cobordisms can be represented by link diagrams in ${\mathbb{R}}^{2}$ and movies between them, there are additional isotopies of link cobordisms in $S^{3}\times{[0,1]}$, that do not exist in ${\mathbb{R}}^{3}\times{[0,1]}$. In addition to the standard Carter–Rieger–Saito movie moves [CS93, CRS97], a link homology theory that is functorial in $S^{3}$ additionally has to satisfy the so-called sweep-around move (1.1), which encodes a small isotopy of a sheet of link cobordism through $\infty\times{[0,1]}$. The central technical result that we prove in §3 is the following.

This move is significantly more complex than any of the Carter–Saito movie moves because it lacks any locality after the projection to ${\mathbb{R}}^{2}$, and thus has to be checked for any tangle $T$ with two endpoints. We do this in §3 and thereby also demonstrate how computable cobordism maps in Khovanov–Rozansky homology have become.