Abstract
We define additional gradings on two generalizations 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 [Formula: see text]-manifolds. In particular, the invariants of embedded surfaces are expressed in terms of certain diagrams related to the thickened [Formula: see text]-manifold, so that we refer to them as picture-valued invariants. This paper contains the first instance of such invariants for [Formula: see text]-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.
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