Stabilization of the Khovanov Homotopy Type of Torus Links

TL;DR

This paper proves that the Khovanov homotopy types of (n,m) torus links stabilize as m approaches infinity, extending known homology stabilization results to homotopy types and providing explicit bounds.

Contribution

It establishes the stable homotopy equivalence of Khovanov homotopy types for torus links as m increases, with explicit bounds and new examples of non-trivial Sq^2 action.

Findings

Khovanov homotopy types stabilize as m→∞

Explicit bounds for stabilization are provided

New examples of torus links with non-trivial Sq^2 action

Abstract

The structure of the Khovanov homology of $(n,m)$ torus links has been extensively studied. In particular, Marko Stosic 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.