Khovanov-Rozansky homologies, Bott-Samelson spaces and twisted cohomology

By means of Rasmussen’s formulation of Khovanov-Rozansky homology originally given over $\mathbb{Q}$ in arXiv:math/0607544, we compare different flavors of $\mathfrak{s}\mathfrak{l}(n)$ link homology with the link invariants obtained by Kitchloo in arXiv:1910.07444 via twistings of Borel equivariant cohomology applied to the symmetry breaking spectra. In particular, we see how these geometric constructions based on Bott-Samelson varieties produce equivariant integral $\mathfrak{s}\mathfrak{l}(n)$ link homology with either specialized or universal potential.