Knot homologies and their deformations

Knot homology theories are powerful generalizations of classical (and quantum) knot polynomials. Besides providing stronger invariants, these theories are often functorial under knot cobordisms and contain additional topological information. I will start by introducing Khovanov homology, a paradigmatic example of a knot homology theory, and explain how it fits into the family of colored sl(N) knot homology theories. The goal of this talk is to explain how deformation techniques help to answer two important questions about this family: What relations exist between its members? What topological information about knots and links do they contain? I will recall Lee's deformation of Khovanov homology and sketch how it generalizes to the case of colored sl(N) link homology. The result is a decomposition theorem for deformed colored sl(N) knot homologies, which leads to new spectral sequences between knot homologies and to new concordance invariants in the spirit of Rasmussen's s-invariant. Part of this is joint work with David E. V. Rose. ​