Stabilization of the Khovanov Homotopy Type of Torus Links

The structure of the Khovanov homology of |$(n,m)$| torus links has been studied extensively. In particular, Marko Stošić proved that the homology groups stabilize as |$m\rightarrow\infty$|⁠. We show that the Khovanov homotopy types of |$(n,m)$| torus links, as constructed by Robert Lipshitz and Sucharit Sarkar, also become stably homotopy equivalent as |$m\rightarrow\infty$|⁠. We provide an explicit bound on values of |$m$| beyond which the stabilization begins. As an application, we give new examples of torus links with non-trivial |$Sq^2$| action.