  We consider compact 3-manifolds M having a submersion h to R in which each
generic point inverse is a planar surface. The standard height function on a
submanifold of the 3-sphere is a motivating example. To (M, h) we associate a
connectivity graph G. For M in the 3-sphere, G is a tree if and only if there
is a Fox reimbedding of M which carries horizontal circles to a complete
collection of complementary meridian circles. On the other hand, if the
connectivity graph of the complement of M is a tree, then there is a
level-preserving reimbedding of M so that its complement is a connected sum of
handlebodies.
  Corollary: The width of a satellite knot is no less than the width of its
pattern knot. In particular, the width of K_1 # K_2 is no less than the maximum
of the widths of K_1 and K_2.
