  For a 3-manifold with torus boundary admitting an appropriate involution, we
show that Khovanov homology provides obstructions to certain exceptional Dehn
fillings. For example, given a strongly invertible knot in S^3, we give
obstructions to lens space surgeries, as well as obstructions to surgeries with
finite fundamental group. These obstructions are based on homological width in
Khovanov homology, and in the case of finite fundamental group depend on a
calculation of the homological width for a family of Montesinos links.
