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

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

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. …