Thompson's group $F$, tangles, and link homology

We extend a construction of Jones to associate $(n,n)$-tangles with elements of Thompson’s group $F$ and prove that it is asymptotically faithful as $n\to \mathrm{\infty }$. Using this construction we show that the oriented Thompson group $\vec{F}$ admits a lax group action on a category of Khovanov’s chain complexes.