Khovanov homology and exotic $4$-manifolds

We show that the Khovanov-Rozansky gl₂ skein lasagna module distinguishes the exotic pair of knot traces X₋₁(-5₂) and X₋₁(P(3,-3,-8)), an example first discovered by Akbulut. This gives the first analysis-free proof of the existence of exotic compact $4$-manifolds. Along the way, we present new explicit calculations of the Khovanov skein lasagna modules, and we define lasagna generalizations of the Lee homology and Rasmussen s-invariant, which are of independent interests. Other consequences of our work include a slice obstruction of knots in $4$-manifolds with nonvanishing skein lasagna module, a sharp shake genus bound for some knots from the lasagna s-invariant, and a construction of induced maps on Khovanov homology for cobordisms in kCP².