Consider a line L in R^3 in the shape of a trefoil knot. Consider the surface S that is the union of all unit circles that have centers on this line and whose tangent vectors are all perpendicular to the tangent vector of L at the cirle's center. S does not intersect itself. 


What is the shortest possible length of L?



Length of shortest possible knot
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 16.372 if we define ropelength using circles of diameter 1. The upper bound is believed to be tighter.


