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If you draw a knot and place lots of evenly-spaced points on it, with straight segments between adjacent points, clearly the knot you started with is the shortest solution to the TSP in 3 dimensions. Question: for a stick knot (straight lines between points) with $v=6$ vertices (the least number of points or segments for a stick knot), is there any shortest TSP path that forms a knot? If not, what is the least such value of $v$?


My question needs clarification. Suppose we have a smooth knot (everywhere a finite curvature). Suppose this knot is "reasonable" such that two separate parts of its length are not closer than $C$. (I don't know how to state that sensibly). If that condition does not hold, just expand the whole knot until it does. Now evenly space points along its length with spacing $\ll C$. This is supposed to assure that the closest points to a given one are its 2 neighbors along the length. Then create a stick knot by connecting each point with its 2 neighbors. It seems clear that the shortest path around the whole stick knot will be the path that just connects each point with its neighbors. Any path that jumps off this ordered set of points will have to travel more than $C$ in both directions. This may be as clear as mud - I can't tell.

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"Tying knot theory"... :D –  Zev Chonoles Jun 5 '12 at 20:21
    
I'm afraid your assertion in the first sentence is not at all clear to me. I'm capable of drawing some pretty zigzaggedy knots. –  Rahul Jun 5 '12 at 20:26
    
Not bad, Rahul ;-), but I also don't get it. –  draks ... Jun 5 '12 at 22:45
    
The least number of segments for a stick knot is 3, not 6, as the unknot is a knot. –  Gerry Myerson Jun 6 '12 at 4:38
    
Now included in a question asked at MO, mathoverflow.net/questions/99213/… –  Gerry Myerson Jun 10 '12 at 6:19
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