Link homology and loop homology

We compute the $k$-colored $\mathfrak{s}\mathfrak{l}(N)$ homology of the torus knot $T(2,2m+1)$, and we show that it stabilizes as $m\to \mathrm{\infty }$ to the integral homology of the free loop space of the complex Grassmannian $\mathrm{G}\mathrm{r}(k,N)$. In particular, when $k=1$ and $N=2$, we observe that the Khovanov homology of $T(2,2m+1)$ stabilizes to the homology of the free loop space of the $2$-sphere.