  The main goal of the present paper is to construct new invariants of knots
with additional structure by adding new gradings to the Khovanov complex. The
ideas given below work in the case of virtual knots, closed braids and some
other cases of knots with additional structure. The source of our additional
grading may be topological or combinatorial; it is axiomatised for many partial
cases. As a byproduct, this leads to a complex which in some cases coincides
(up to grading renormalisation) with the usual Khovanov complex and in some
other cases with the Lee-Rasmussen complex.
  The grading we are going to construct behaves well with respect to some
generalisations of the Khovanov homology, e.g., Frobenius extensions. These new
homology theories give sharper estimates for some knot characteristics, such as
minimal crossing number, atom genus, slice genus, etc.
  Our gradings generate a natural filtration on the usual Khovanov complex.
There exists a spectral sequence starting with our homology and converging to
the usual Khovanov homology.
