Twisting bordered Khovanov homology

We describe a bordered version of totally twisted Khovanov homology. We first twist Roberts’s type $D$ structure by adding a “vertical” type $D$ structure which generalizes the vertical map in twisted tangle homology. One of the distinct advantages of our type $D$ structure is that it is homotopy equivalent to a type $D$ structure supported on “spanning tree” generators. We also describe how to twist Roberts’s type $A$ structure for a left tangle in such a way that pairing our type $A$ and type $D$ structures will result in the totally twisted Khovanov homology.