Mini cours

Christian Blanchet,Université de Bretagne Sud

Khovanov homology

Abstract

Khovanov's construction associates to a link diagram a bigraded (co)chain complex. The homology of this complex is a link invariant: the Khovanov homology. Its graded Euler characteristic is equal to the Jones polynomial; this is called a categorification of the Jones polynomial.

We will present a construction of Khovanov's complex and of the homotopy equivalences corresponding to Reidemeister moves. We will introduce the version which allows us to study the slice genus of knots, and Rasmussen's prove of Milnor's conjecture on the slice genus of torus knots will be given. Finally, we will examine constructions for other quantum invariants: sl(3) Khovanov theory and Khovanov-Rozansky sl(N) theory. We will finish with a new construction of a categorification of sl(N) quantum invariant based on trivalent TQFT.