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Grid Movies

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.