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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 todefine invariants of various kinds of embeddings. These include invariants oflinks in thickened surfaces and of surfaces embedded in thickened$3$-manifolds. In particular, the invariants of embedded surfaces are expressedin terms of certain diagrams related to the thickened $3$-manifold, so that werefer to them as picture-valued invariants. This paper contains the firstinstance of such invariants for $2$-dimensional objects.  The additional gradings are defined using cohomological and homotopicinformation of surfaces: using this information we decorate the smoothings ofthe standard Khovanov cube, before transferring the decorations into algebra.

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