  We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie
theorem. Specifically, we give definitions for grid movies, grid movie
isotopies and present a definition of grid planar isotopy as a particular
subset of the grid diagram moves: stabilization, destabilization and
commutation. We show that grid planar isotopy classes are in 1-1 correspondence
with smooth planar isotopy classes by using a new planar grid algorithm that
takes a smooth knot diagram to a grid diagram. We then present generalizations
of both the smooth and grid movie theorems that apply to surfaces with
boundary.
