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

TL;DR

This paper compares various geometric and algebraic approaches to $ ext{sl}(n)$ link homology, revealing how Bott-Samelson varieties and twisted cohomology relate to Khovanov-Rozansky invariants, bridging topology and geometry.

Contribution

It establishes a connection between geometric constructions using Bott-Samelson varieties and algebraic link homologies, providing a unified perspective on $ ext{sl}(n)$ invariants.

Findings

Different flavors of $ ext{sl}(n)$ link homology are compared.

Geometric constructions produce equivariant integral $ ext{sl}(n)$ link homology.

Relations between Khovanov-Rozansky homology and Borel equivariant cohomology are clarified.

Abstract

By means of Rasmussen's formulation of Khovanov-Rozansky homology originally given over $\mathbb{Q}$ in arXiv:math/0607544, we compare different flavors of $\mathfrak{sl}(n)$ 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 $\mathfrak{sl}(n)$ link homology with either specialized or universal potential.