ScalingStacks

Remark 3.3. In some cases it is known that the braid group actions on the link homology of cabled links factor through the symmetric group. For Khovanov homology of links in $\mathbb{R}^{3}$, this was shown by Grigsby–Licata–Wehrli [12, Theorem 2]. For the $\mathfrak{gl}_{N}$ homology of links in $\mathbb{R}^{3}$ (or $S^{3}$) a similar argument works in the case of parallelly oriented strands [11, Section 6.1]. We have no reason to doubt that the same could be true for anti-parallel strands, i.e. in the situation relevant for $\mathcal{S}_{0}^{N}$, but we do not currently know how to prove it.