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.