A link is a (finite) collection of circles, possibly knotted and linked, in the 3-dimensional space. The study of links is a central topic in low-dimensional topology. An important tool to study knots and links is given by link homology theories. Khovanov introduced one of these theories in the early 2000s. A few years later, together with Rozansky, Khovanov defined a family of homology theories called Khovanov-Rozansky homologies.

The aim of these lectures is to provide an introduction to Khovanov-Rozansky homologies covering the basic definitions, important properties, several variants, and some applications. We chose to focus mostly on topological applications and, in particular, applications to the study of concordance.

The plan is to quickly cover some basic material on knot theory, and then start with Khovanov homology. We will introduce the

-invariant and see the main applications of this theory. Then, we will turn to the definition of the Khovanov-Rozansky homologies. We focus on two different definitions: the original via matrix factorizations and the definition via foams. Afterwards, an idea of how to prove functoriality for Khovanov-Rozansky homologies is given. We conclude with an overview of the applications of these theories and open problems.

Time permitting, in the final part of the course, we will cover some additional topics which may depend on the students’ interests.