  Our main result has topological, combinatorial and computational flavor. It
is motivated by a fundamental conjecture stating that computing Khovanov
homology of a closed braid of fixed number of strands has polynomial time
complexity. We show that the independence simplicial complex $I(w)$ associated
to the 4-braid diagram $w$ (and therefore its Khovanov spectrum at extreme
quantum degree) is contractible or homotopy equivalent to either a sphere, or a
wedge of 2 spheres (possibly of different dimensions), or a wedge of 3 spheres
(at least two of them of the same dimension), or a wedge of 4 spheres (at least
three of them of the same dimension). On the algorithmic side we prove that
finding the homotopy type of $I(w)$ can be done in polynomial time with respect
to the number of crossings in $w$. In particular, we prove the wedge of spheres
conjecture for circle graphs obtained from 4-braid diagrams. We also introduce
the concept of Khovanov adequate diagram and discuss criteria for a link to
have a Khovanov adequate braid diagram with at most 4 strands.
