Knotted Surfaces in 4D

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Knotted Surfaces in 4D

An important special case of surfaces in 4D is the subject of knotted surfaces. While closed curves are knottable in 3D, smooth curves (whether or not they are thickened) can always be untied without self intersection in 4D. However, surfaces can be knotted in 4D. Some surfaces in 4D appear to be knotted but are really unknotted. They can be ``untied'' in principle by a series of deformations developed by Dennis Roseman during which the surface does not develop self-intersections; such deformations are examples of isotopies.

The important topological as well as graphical problem is that of determining which a priori characteristics of an apparently knotted surface guarantee that it is isotopic to another surface; of particular interest is determining whether a surface is isotopic to an embedded sphere, and thus unknotted.

Examples of strategies for understanding these issues range from analyzing 4D slices of the surface and projecting them to 3D, as shown in Figure 6, to showing cutaway interiors or providing above-below crossing markings on the self-intersections of the 3D projection, as shown in Figure 7. The latter may be thought of as a special case of the color coded 4D depth method in Figure 1; Roseman has also experimented with the use of varying 4D-depth-keyed texture sizes to appeal to ``near'' and ``far'' visual preconceptions, illustrated in Figure 6. In the next section, we note a method involving thickened surfaces that automatically provides occlusion cues on the 3D image of the 4D structure; these occlusions are analogous to the 3D occlusion cues observed in 2D images of curves thickened into tubes.

Tamara Munzner
Thu Sep 21 19:17:33 CDT 1995