Geometry & Topology Seminars

Khovanov's categorification of the Jones polynomial assigns a chain complex to a knot in the 3-sphere. The homology groups of the chain complex are also an invariant of the knot called Khovanov homology. Through Cooper-Krushkal's categorification of the Jones-Wenzl projector, one also obtains a categorification of the colored Jones polynomial. Due to the recent applications to Khovanov homology to low-dimensional topology, it is natural to ask how it could apply to the study of the relationship between the colored Jones polynomial and hyperbolic structures on the knot complement. I will introduce colored Khovanov homology and discuss how the study of its structure may help reconcile the algebraic aspects of the colored Jones polynomial to its geometry.