Additional gradings on generalisations of Khovanov homology and invariants of embedded surfaces

Abstract

We define additional gradings on two generalisations of Khovanov homology (one due to the first author, the other due to the second), and use them to define invariants of various kinds of embeddings. These include invariants of links in thickened surfaces and of surfaces embedded in thickened $3$-manifolds. In particular, the invariants of embedded surfaces are expressed in terms of certain diagrams related to the thickened $3$-manifold, so that we refer to them as picture-valued invariants. This paper contains the first instance of such invariants for $2$-dimensional objects. The additional gradings are defined using cohomological and homotopic information of surfaces: using this information we decorate the smoothings of the standard Khovanov cube, before transferring the decorations into algebra.