Some Combinatorial Aspects of Movies and Movie-moves in the Theory of Smoothly Knotted Surfaces in IR

Throughout this paper, by a knotted surface (or, synonymously, 2-knot) will always be meant a smooth embedding in IR of a compact 2-manifold M (where M is smooth, not necessarily connected, and is not assigned an orientation.) 1 By isotopy of two such smoothly knotted surfaces in IR, will be meant smooth ambient isotopy. There is a beautiful combinatorial description of the isotopy-classes of such smooth knotted surfaces in IR, via the ”doubly time-oriented movies” of Carter, Rieger and Saito ([CRS]), explained below in § 2.3. Putting all this in a more general context, the collection of smooth isotopy classes of knotted surfaces is a portion of the collection of “two-tangles”, whose intricate structure (that of a ”free braided monoidal 2-category with duals”)has been elucidated by the work of Fischer ([F]), Kharlamov and Turaev([KT]) Baez and Neuchl ([BN]) and especially in the recent work of Baez and Lang-