Representation stability in link homology

Question: Does Khovanov homology exhibit representation stability with respect to cabling?

Given a knot $K$ in the 3-sphere, we denote by $$\mathit{Kh}^{i,j}(K)$$ its Khovanov homology in homological grading $i$ and quantum grading $j$. Let $K_{n}$ be the $n$-cable of the knot. Grigsby-Licata-Wehrli associate to each braid $\beta \in B_n$, a link cobordism in $S^3 \times I$, $$\Sigma_\beta: K_{n} \to K_{n},$$ and prove that the induced map on Khovanov homology depends only on the permutation associated with $\beta$. In this way, each element $\sigma \in S_n$ gives rise to an automorphism $$\Phi_\sigma: \mathit{Kh}^{i,j}(K_n) \to \mathit{Kh}^{i,j}(K_n),$$ which turns the Khovanov homology of the $n$-cable in each bigrading into an $S_n$-representation. The dimensions of the vector spaces $\mathit{Kh}^{i,j}(K_n)$ are unbounded as $n \to \infty$. But the question is asking whether the decomposition of $\mathit{Kh}^{i,j}(K_n)$ into irreducible $S_n$-representations stabilizes as $n \to \infty$, in the sense of Church-Farb.

Church-Farb first discovered this "representation stability" phenomenon in the cohomology of the configuration spaces $\mathrm{Conf}_n(\mathbb{C})$ of $n$ points in the complex plane as $n \to \infty$. Representation stability turns out to be quite ubiquitous and has been an important area of exploration over the last decade, but it does not yet seem to have been explored in the setting of link homology theories (Khovanov homology, knot Floer homology).

Church-Ellenberg-Farb introduced the notion of FI-modules in order to unify and explain the various representation stability phenomena observed in nature. In brief, an FI-module $V$ is a sequence of vector spaces with linear maps between them, $$\dots \to V_m \to V_{m+1} \to V_{m+2} \to \dots$$ where each $V_m$ is an $S_m$-representation and the maps between these vector spaces are equivariant in a natural sense. This FI-module is said to be finitely generated if there is a finite set $S \subset \sqcup_m V_m$ such that no proper sub-FI-module of $V$ contains $S$. They then prove that the vector spaces $V_m$ in the FI-module form a representation stable sequence if and only if $V$ is finitely generated.

This suggests an approach to answering the question at the top. In particular, there are natural link cobordisms which induce maps $$\mathit{Kh}^{i,j}(K_n) \to \mathit{Kh}^{i,j}(K_{n+2}),$$ which are appropriately equivariant with respect the symmetric group actions on either side. It should be a tractable problem to decide whether the FI-module given by $$V_m = \mathit{Kh}^{i,j}(K_{2m})$$ is finitely generated.

Finally, we remark that the question at the top may have relevance for understanding the structure of the ($N=2$) skein lasagna module of Morrison-Walker-Wedrich, which assigns a group to a 4-manifold with a (possibly empty) link in its boundary. Manolescu-Neithalath proved that the skein module of a 2-handlebody, obtained from the 4-ball by attaching 2-handles along a link in the 3-sphere, can be interpreted via the Khovanov homology groups of cables of the link components (though the strands in these cables are not oriented coherently), modulo relations involving the braid group action above as well as (versions of) the maps $\mathit{Kh}(K_n) \to \mathit{Kh}(K_{n+2})$ above.

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