The invariant you are talking about is usually called the "ropelength" of the knot. You can find some basic stuff at the wikpedia page http://en.wikipedia.org/wiki/Ropelength which also gives some good references. (Note that some people use unit circles, while other people use circles of diameter 1, so the reported ropelength differs by a factor of 2.)
The exact value of the ropelength is not known for any nontrivial knot. However in the case of the trefoil, there are some pretty good bounds. It is between 15.66 and 16.34 16.372 if we define ropelength using circles of diameter 1. The upper bound is believed to be tighter.
