TITLE: Visualising locally flat embeddings of surfaces in R^4
QUESTION [10 upvotes]: As far as I understand it follows from the work of M. Freedman that there exist  locally flat embeddings of two dimensional surfaces in $\mathbb R^4$ that can not be smoothed in the class of locally flat embedding. (For definition of locally flat see http://en.wikipedia.org/wiki/Local_flatness)
I am curious if one can draw a realistic picture of such a surface. At least is it possible to draw an intersection of such a surface with a (linear) hyperplane in $\mathbb R^4$?
Is the Hausdorff dimension of such an intersection equals $1$?

REPLY [3 votes]: I have been thinking about this as well. My approach would be to construct a broken surface diagram or chart of an immersed disk that a classical knot bounds. For simplicity take the untwisted double of the Figure-8 knot ($4_1$) or the Conway knot (11 crossings in the table). Take any immersed disk bounded by the knot with transverse double points. Now find a pair of arcs that the double points bound and look for an immersed disk bounded by these. In my opinion, you should be able to get one or two levels down before the figure becomes impossible to read. The tricks, though, are to use a Hoffman coding for the depths of the folds and double points and draw the lines with differing colors and line thicknesses. 
This project is on my (rather long) to-do list. If I make progress, I'll let you know.