Categorification of Quantum Invariants and the Slice-Bennequin Inequality in Contact Topology

This monograph explores the intersection of quantum topology and contact geometry, specifically focusing on the categorification of the Jones polynomial via Khovanov homology and its application to the Slice-Bennequin inequality. We analyze the construction of Rasmussen's s-invariant, a concordance homomorphism derived from the spectral sequence of Lee's deformation of the Khovanov complex. By examining Plamenevskaya's transverse invariant (K), a distinguished class in Khovanov homology, we provide a rigorous derivation of the bound sl(K) s(K) - 1 for transverse knots in the standard contact 3-sphere. This result establishes a purely combinatorial proof of the Slice-Bennequin inequality, sl(K) 2g4(K) - 1, connecting the self-linking number of a transverse knot to the smooth slice genus. We further discuss the implications of these bounds for the existence of exotic smooth structures on 4-manifolds and the non-sliceness of certain quasipositive knots.