Though a knotted curve can obviously be made as short as we like, a certain length is required to actually tie a physical rope in a knot. As a mathematical idealization of this situation, we consider the problem of minimizing the length of a knot (or braid, or link) subject to the condition that its normal disks of radius 1 do not intersect. This problem has been studied by Gonzalez-Maddocks-Schuricht-von der Mosel, Cantarella-Kusner-Sullivan and others, but the actual shortest length of a knot remains unknown.

Since the constraint in this problem is nonholonomic, there is no Euler-Lagrange equation in the usual sense. However we present a measure-theoretic substitute that is adequate for many purposes, and use it to analyze some curious examples of ropelength critical points. Moreover this approach is based on an analogy with a certain family of nonholonomic minimization problems in finite dimensions, which is the basis for further conjectures. 

This is joint work with J. Cantarella, R. Kusner, J. Sullivan and N. Wrinkle.