  We modify an approach of Johnson to define the distance of a bridge splitting
of a knot in a 3-manifold using the dual curve complex and pants complex of the
bridge surface. This distance can be used to determine a complexity, which
becomes constant after a sufficient number of stabilizations and perturbations,
yielding an invariant of the manifold-knot pair. We also give evidence toward
the relationship between the pants distance of a bridge splitting and the
hyperbolic volume of the exterior of a knot.
