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Hey,

The carpenter's ruler problem is about polygons you can make with planar joints. You can formulate a similar question for knots when you introduce ball joints:

Given m ball joints connected by rods of arbitrary length, which knots can you make?

I found out that you can make the first non trivial knot with your fingers and thumbs and your thumbs are ball joints. This is probably the same question as for stick knots and since the stick number is not known for all stick knots, the answer to the question I must say is, no, the question is an open question.

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Yes, in general stick numbers for knots are unknown, although there is an upper bound: at most twice the crossing number. Here is table of known stick numbers: colab.sfu.ca/KnotPlot/sticknumbers. (I had earlier asked an MO question about stick numbers and crossing numbers: mathoverflow.net/questions/39870). –  Joseph O'Rourke Apr 26 '11 at 0:18
    
The standard question for knots is :Given a knot G, what is the stick number for G (and the stick number is equal to the joint number)? My question is in the reverse sense:Given n sticks, what knot classes can one make? This comes down to: Given m crossings (m=n/2 where n is stick number), which knot classes can be created? Is this an open question? If this is not open and instead is a known theorem, then we can give all the knots which can be created with n joints... –  Ben Sprott Apr 26 '11 at 19:14
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In general this is open: "Given $n$ sticks, what knot classes can one make?" For a specific version: Given $n=c+2$ sticks, can you make a 2-braid with $c$ crossings? That is open; $c+3$ suffice, but it is unknown if $c+3$ are needed. (Here I am relying on work of Colin Adams.) –  Joseph O'Rourke Apr 26 '11 at 20:26
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