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.

Thu Sep 21 19:17:33 CDT 1995