A Flip-book of Knot Diagrams for Visualizing Surfaces in 4-Space

Abstract

Just as 2D shadows of 3D curves lose structure where lines cross, 3D graphics projections of smooth 4D topological surfaces are interrupted where one surface intersects itself. They twist, turn, and fold back on themselves, leaving important but hidden features behind the surface sheets. In this paper, we propose a smart slicing tool that can read the 4D surface in its entropy map and suggest the optimal way to generate cross-sectional images - or ''slices'' - of the surface to visualize its underlying 4D structure. Our visualization thinks of a 4D-embedded surface as a collection of 3D curves stacked in time, very much like a flip-book animation, where successive terms in the sequence differ at most by a critical change. This novel method can generate topologically meaningful visualization to depict complex and unfamiliar 4D surfaces, with the minimum number of cross-sectional diagrams. Our approach has been successfully used to create flip-books of diagrams to visualize a range of known 4D surfaces. In this preliminary study, our results show that the new visualization and slicing tool can help the viewers to understand and describe the complex spatial relationships and overall structures of 4D surfaces.