Abstract : One of the main goal of Knot Theory is to develop tools allowing to tell apart different knots. Namely, one wants to tell if two embedding K and K′ of the
circle \( S^1\) into \( R^3\) can be related by an ambient isotopy. Numerous invariants of knots have been developed since the 19th century and among them polynomial invariants form
an important class. The Jones polynomial V (K) introduced in 1984 can be computed directly from a diagram of the knot K, that is, a (regular) projection of the knot K onto the plane \( R^2\). In 1999 Khovanov introduced a new invariant, nowadays called Khovanov homology, that refines the Jones polynomial. Given a diagram D of a knot K, one can compute a complex C(D) of modules over an algebra A. The homology groups \(H^i(C(D))\) then form an invariant of K and one can recover the Jones polynomial by taking the
alternating sum of the Euler characteristic of these groups. This is a particular instance of a categorification.
The goal of this presentation will be to understand the construction of the Khovanov homology of a knot diagram.