A Combinatorial Description of Knotted Surfaces and Their Isotopies

We discuss the diagrammatic theory of knot isotopies in dimension 4. We project a knotted surface to a three-dimensional space and arrange the surface to have generic singularities upon further projection to a plane. We examine the singularities in this plane as an isotopy is performed, and give a finite set of local moves to the singular set that can be used to connect any two isotopic knottings. We show how the notion of projections of isotopies can be used to give a combinatoric description of knotted surfaces that is sufficient for categorical applications. In this description, knotted surfaces are presented as sequences of words in symbols, and there is a complete list of moves among such sequences that relate the symbolic representations of isotopic knotted surfaces.