# Length of shortest possible knot

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?

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To add a remark to Oliver's answer: Until the 2006 paper "Quadrisecants Give New Lower Bounds for the Ropelength of a Knot" (arXiv math.DG/0408026), it was unknown if a foot-long rope 1 inch in diameter could tie a trefoil. Their lower bound of 15.66 showed the answer is: No! – Joseph O'Rourke Jan 31 '11 at 1:38
@Joseph I believe the answer to the foot-long root question was actually resolved a few years earlier in Y Diao, The lower bounds of the lengths of thick knots, J. Knot Theory Ramiﬁcations 12 (2003) math.uncc.edu/files/preprint/2002/2002_06.pdf – Oliver Jan 31 '11 at 5:33
Oliver: I stand corrected! Thanks. – Joseph O'Rourke Jan 31 '11 at 11:36