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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 $1$?

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    $\begingroup$ Not quite what you want, but if you take a knot in $R^3$ which is topologically slice but not slice, e.g. the whitehead double of the trefoil, it bounds a locally flat continuously embedded disk in upper half-$R^4$, but does not bound a smooth such disk. So the intersection of this weird disk with a linear hyperplane is a smooth knot. $\endgroup$
    – Paul
    Commented Nov 28, 2012 at 15:54
  • $\begingroup$ Paul I understand that some of intersections of the surface with linear subspaces can be smooth, as in you example. I wonder if one can tell what would be the worst possible intersection for your example. How would it look like? $\endgroup$
    – aglearner
    Commented Nov 29, 2012 at 15:49
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    $\begingroup$ Coming at this with a lot of ignorance, this raises the following related question: are there locally-flat embeddings of surfaces in $\mathbb R^4$ that are not smoothable embeddings but such that the intersection with every hyperplane parallel to a given one are all sufficiently "nice" (either smoothly embedded curves, smoothly immersed curves, or finite sets of points (or unions of such things))? $\endgroup$
    – Greg Friedman
    Commented Dec 5, 2012 at 6:12

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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.

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