ScalingStacks

Current constructions of Khovanov–Rozansky link homologies proceed via a functorial invariant of tangles and tangle cobordisms up to isotopy, taking values in the bounded homotopy category of an additive category; see §2. Our proof of Theorem 1.1 is stronger than necessary in the sense that it shows that a certain equivalent reformulation of the sweep-around move holds on the chain level (i.e. not just up to homotopy) provided the tangle $T$ is presented as a partial braid closure.

It is an open question whether the Khovanov–Rozansky homologies are truncations of homotopy-coherent versions with values in chain complexes over the same additive category. If this is indeed the case, then it is plausible that our method of proof would be suitable for an analog of Theorem 1.1 in this setting. Given a fully homotopy-coherent invariant of links in $S^{3}$, we could construct a disklike 4-category enriched in chain complexes (rather than abelian groups), and then use a homotopy colimit construction to extend this invariant to 4-manifolds [MW12, §7]. The result would be a well-defined-up-to-coherent-homotopy chain complex assigned to a 4-manifold $W$ and a boundary condition $L$. At the end of this process we could take homology of this chain complex to produce an abelian group. The invariants $\mathcal{S}^{N}_{*}(W;L)$ can be thought of as an approximation to the latter, given by taking homology (too) early in the construction. One would then expect the two theories to be related by a spectral sequence.