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This is a refinement of one of my earlier questions (I failed to put into words what I really wanted to ask). First of all, I'm not sure "singularity" is the correct word to use hence the quotes. Consider the following wild knot:

enter image description here

Then what exactly happens where the curls get infinitely small? Is the knot still differentiable there? I'm asking because I'm trying to understand why requiring a knot to be differentiable is not enough to prevent knots from being wild. On the other hand, smoothness is enough.

Thanks for help!

(If anyone knows the parametric equation of this curve it might make it easier to see what happens.)

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    May I ask how you draw that beautiful picture? What software did you use? How did you insert it here?2012-10-24
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    @HuiYu I downloaded the picture from the [Wikipedia article about wild knots](http://en.wikipedia.org/wiki/Wild_knot). Then I uploaded it using the upload image button (you can see the button if you edit one of your posts).2012-10-24

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