Thin position for knots in a 3-manifold
  We extend the notion of thin multiple Heegaard splittings of a link in a 3-manifold to take into consideration not only compressing disks but also cut-disks for the Heegaard surfaces. We prove that if H is a c-strongly compressible bridge surface for a link K contained in a closed orientable irreducible 3-manifold M then one of the following is satisfied:   H is stabilized,   H is meridionally stabilized,   H is perturbed, a component of K is removable, M contains an essential meridional surface. 
Maggy Tomova