An gentle introduction to Khovanov–Rozansky homology

Classical quantum topology studies polynomial invariants of links arising from the representation of quantum groups — in particular, the sln-Jones polynomial, associated to quantum sln. Around 2000, it was realized that some of these polynomial invariants generalize (“categorify”) to homological invariants — in particular, Khovanov–Rozansky homology categorifying the sln-Jones polynomial. This homology can be defined in various ways, including using matrix factorisation, category O, the algebraic geometry of flag varieties, or defect TQFTs. String diagrams for higher categories, i.e. stratified surfaces, naturally appear.

This talk will survey the main idea behind the construction of Khovanov–Rozansky homology, common to all approaches. We will then describe a combinatorial definition of Khovanov–Rozansky homology via the foam evaluation formula due to Robert and Wagner. Since it is combinatorial, there are no prerequisite, and it gives a common ground for people coming from different background.