State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors (Quandle cohomology and state-sum invariants of knotted curves and surfaces, preprint). In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with Maple programs are used to evaluate the invariants for knot table. For knotted surfaces in 4-space, movie methods and surface braid theory are used. Relations between the invariants and symmetries of knots are discussed. Academic Press

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.1997 Academic Press

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Using the thin presentation of an n-gem, for n=3, 4, we prove that a set of local moves for crystallizations are sufficient in the sense that any two crystallizations inducing the same n-manifold are related by a finite sequence of moves taken from this set. In dimension 3, we apply the moves to prove an identity among quantum 6j-symbols, and we propose a new model for a quantum topological invariant. In dimension 4, we observe that the thin presentation reduces the description of a 4-manifold to a 3-dimensional one. Academic Press