  If a knot K in a closed, orientable 3-manifold M has a bridge surface T with
distance at least 3 in the curve complex of T - K, then the genus of any
essential surface in its exterior with non-empty, non-meridional boundary gives
rise to an upper bound for the bridge number of K with respect to T. In
particular, a nontrivial, aspherical, and atoroidal knot K with such a bridge
surface has its bridge number bounded by 5 if K has a non-trivial reducing
surgery; 6 if K has a non-trivial toroidal surgery; and 4g + 2 if K is
null-homologous and has Seifert genus g.
