Link homology and loop homology

TL;DR

This paper computes the colored -homology of certain torus knots and shows it stabilizes to the homology of free loop spaces of complex Grassmannians, revealing deep connections between knot theory and algebraic topology.

Contribution

It establishes a new link between knot homology and the homology of free loop spaces of Grassmannians, demonstrating stabilization phenomena.

Findings

Colored -homology stabilizes as m 0f0 m 0f0 0f0f0 to the homology of free loop spaces.

For k=1, N=2, Khovanov homology stabilizes to the homology of the free loop space of the 2-sphere.

Shows a connection between knot invariants and algebraic topology of Grassmannians.

Abstract

We compute the $k$-colored $\mathfrak{sl}(N)$ homology of the torus knot $T(2,2m+1)$, and we show that it stabilizes as $m\to\infty$ to the integral homology of the free loop space of the complex Grassmannian $\mathrm{Gr}(k,N)$. 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.