From Khovanov homology to the Rasmussen s-invariant via Lee homology

This thesis traces a path from the construction of Khovanov homology to the definition of the Rasmussen s-invariant via Lee homology. After reviewing the foundational theory of knots and links and introducing the Jones polynomial through the Kauffman bracket, we develop Khovanov homology as a categorification of the Jones polynomial and establish its invariance under Reidemeister moves. We then study Lee homology, a deformation of Khovanov homology, and construct the spectral sequence converging from Khovanov homology to Lee homology. This structure gives rise to the Rasmussen s-invariant, whose key properties are established and applied to obtain a sharp lower bound on the slice genus, yielding in particular a combinatorial proof of the Milnor conjecture for torus knots.