Topology Seminar

The Dold-Kan constructon in simplicial homotopy theory can be applied to convert link homology theories into homotopy theories. We construct a mapping  F: L -----> S taking link diagrams L to a category of simplicial spaces S such that up to looping or delooping, link diagrams related by Reidemeister moves will give rise to homotopy equivalent  simplicial objects, and the homotopy groups of these objects will be equal to the link homolgy groups of the original link homology theory. The construction is independent of the particular link homology theory, applying equally well to Knovanov Homology and to Knot Floer Homology and other theories of these types. The construction is of particular interest for Khovanov Homology where there is a natural pre-simplicial structure already present in the definition of the Khovanov category. This allows us to define an embedding of the cube category into a simplicial category so that the map to Frobenius algebras determined by a knot or link produces a simplicial module. The homology of this simplicial module is Khovanov homology. The homotopy type of this simplicial module is the homotopy type to which we refer above.