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Imagine you take a knot or link, and "fatten" it as much as possible, so that the surface is snug against itself at several places. Then "fuse" it into a solid object. What I have in mind is something like this:
  Fattened Knots and Links
The linked tori at the left (image from an MO question) fuse to an object of genus 0. The ideal trefoil (center, image from Wikipedia) I want to say has genus 1 when maximally fattened and fused. The knot $6_2$ to the right (image from Jason Cantarella's site) has, I think, genus 4 or 5, if fully fattened (more than shown). I would want to say the unknot fattens to genus 1, never entirely closing. I have two questions:

Q1. Can this notion of fused, fattened knots be formalized consistently? It seems that fusing should occur where surface curvatures match, whereas, say, at the center of the ideal trefoil, the curvatures are opposed.

Q2. Is there a measure on knots that corresponds, or yields this fattened genus?

This is a blue-sky exploratory well, so to speak.

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Beautiful pictures, as always! I wonder though if this might depend on the embedding? For example, an unknot that is simply a circle will thicken to a genus 1 solid object (using your convention), but I don't see immediately that this will be true of any unknot. – Zev Chonoles Feb 8 '12 at 2:11
@Zev: Yes, you are right, this depends quite on the embedding. Good point! – Joseph O'Rourke Feb 8 '12 at 2:48
Maybe it is possible to achieve a non-trivial minimum for the fat genus of any embedding of a certain knot – Dilitante Feb 8 '12 at 8:45
@James: That's what I'm hoping! – Joseph O'Rourke Feb 8 '12 at 12:37

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