Representation stability in link homology

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

Given a framed 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. 

Now, the dimensions of the Khovanov homologies $\mathit{Kh}^{i,j}(K_n)$ are unbounded as $n \to \infty$, but one could take the question above to be asking whether, despite this unboundedness in dimension, 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. However, this naive version of the question does not hold even for the 0-framed unknot $U$. Indeed, the dimension of $\mathit{Kh}^{0,0}(U_{2n})$ is given by $2n \choose n$, which grows exponentially in $n$, whereas Church-Farb proved that representation stability would imply polynomial growth (thanks to David Treumann for pointing this out). Still, one could ask whether for each $i$ and $j$ the groups $$\mathit{Kh}^{i-f(n), \,j-g(n)}(K_n)$$ stabilize as $S_n$-representations, after shifting the bigrading by some functions $f$ and $g$ depending only on $n$.

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. An FI-module $V$ is, roughly, 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 is actually what's called a consistent sequence in Church-Farb; these can be promoted to FI-modules under a mild additional hypothesis). An FI-module is 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.

It's not clear how to turn the sequence of Khovanov homologies $\mathit{Kh}^{i,j}(K_n)$ into an FI-module. On the other hand, there is a natural dotted link cobordism from $K_n$ to $K_{n+2}$ which introduces two new oppositely oriented strands and induces a map $$\mathit{Kh}^{i,j}(K_n) \to \mathit{Kh}^{i,j-2}(K_{n+2}).$$ This suggests the definition of an FI-module with $$V_m = \mathit{Kh}^{i,j-2m}(K_{2m}),$$ where $K_{2m}$ now stands for the link obtained from the $m$-cable by replacing each strand with two oppositely oriented strands, and $S_m$ acts on $\mathit{Kh}^{i,j}(K_{2m})$ in the obvious way (by permutations of the solid tori containing paired oppositely oriented strands). One can then ask whether this FI-module is finitely generated.

It would also be interesting, and perhaps more tractable, to explore these questions in the setting of knot Floer homology. The advantage there is that one can recover the Floer homology of $K_n$ from the bordered Floer invariants of 1) the complement of $K$ and 2) the $n$-cable of the core in the solid torus, thanks to the work of Lipshitz-Ozsvath-Thurston. This localizes the problem to understanding how the invariant of the cable in the solid torus changes with $n$. There are natural braid group actions in this setting, but it is not clear that these descend to $S_n$-actions (and is probably not true depending on the version of knot Floer homology one considers).

Finally, we remark that these kinds of questions are loosely related to 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, modulo relations involving the braid group action above as well as the maps $\mathit{Kh}(K_n) \to \mathit{Kh}(K_{n+2})$ above.

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