  It was proven by Gonz\'alez-Meneses, Manch\'on and Silvero that the extreme
Khovanov homology of a link diagram is isomorphic to the reduced (co)homology
of the independence simplicial complex obtained from a bipartite circle graph
constructed from the diagram. In this paper we conjecture that this simplicial
complex is always homotopy equivalent to a wedge of spheres. In particular, its
homotopy type, if not contractible, would be a link invariant and it would
imply that the extreme Khovanov homology of any link diagram does not contain
torsion. We prove the conjecture in many special cases and find it convincing
to generalize it to every circle graph (intersection graph of chords in a
circle). In particular, we prove it for the families of cactus, outerplanar,
permutation and non-nested graphs. Conversely, we also give a method for
constructing a permutation graph whose independence simplicial complex is
homotopy equivalent to any given finite wedge of spheres. We also present some
combinatorial results on the homotopy type of finite simplicial complexes and a
theorem generalizing previous results by Csorba, Nagel and Reiner, Jonsson and
Barmak. We study the implications of our results to Knot Theory; more
precisely, we compute the real-extreme Khovanov homology of torus links
$T(3,q)$ and obtain examples of $H$-thick knots whose extreme Khovanov homology
groups are separated either by one or two gaps as long as desired.
