Informally, the ropelength of a knot in R^3 is the minimal length of rope of thickness 1 that is required to build the knot out of rope. Making this precise is a bit technical, but here is a summary of the approach developed by Gonzalez and Maddocks. (See [1] and [3].)
For distinct points x, y, and z in R^3, let t(x,y,z) denote the radius of the circle that contains all three points. (This is infinite if the point are collinear.) Then the thickness of an embedded circle J in R^3, denoted t(J), is defined to be the infimum of t(x,y,z) for all triples (x,y,z) of points on J. The ropelength of a knot K is the infimum of t(J) for all J isotopic to K.
So far, only estimates of ropelengths, that is numerically computed upper bounds, are known for knots, and many of these are very far from any theoretical lower bound. The intial values of ropelengths presented in KnotInfo were found by Ashton-Cantarella-Piatek-Rawdon and by Klotz-Anderson. We also mention earlier computations performed by Brian Gilbert [] that were not used in compiling the data for KnotInfo
Thanks go to Alex Klotz and Jason Cantarella for assistance in adding ropelength to KnotInfo.
[1] Ted Ashton, Jason Cantarella, Michael Piatek, and Eric Rawdon. Knot tightening by constrained gradient descent. J. Exp. Math. 20 (2011), no. 1, 57--90.
[2] Brian Gilbert. Ideal knots. Posted on The Knot Atlas.
[3] Oscar Gonzalez and John H. Maddocks. Global curvature, thickness, and the ideal shapes of knots. Proc. Nat. Acad. Sci. (USA) 96 (1999), 4769-4773.
[4] Alexander R. Klotz and Caleb J. Anderson, Ropelength and writhe quantization of 12-crossing knots, arxiv posting.
[5] T. Kuriya and O. Shehab. The Lomonaco-Kauffman Conjecture; J. Knot Theory Ramif. 2014, Vol. 23, Issue 1.
[6] Robert B. Kusner and John M. Sullivan. On distortion and thickness of knots. Topology and geometry in polymer science (Minneapolis, MN, 1996), 67--78, IMA Vol. Math. Appl., 103, Springer, New York, 1998.