The
ropelength problem asks for the minimum-length configuration of a knotted
diameter-one tube embedded in Euclidean three-space. The core curve of such a tube
is called a tight knot, and its length is a knot invariant measuring complexity. In
terms of the core curve, the thickness constraint has two parts: an upper bound on
curvature and a self-contact condition.
We give a set of necessary and sufficient conditions for criticality with respect to this
constraint, based on a version of the Kuhn–Tucker theorem that we established in previous
work. The key technical difficulty is to compute the derivative of thickness under a
smooth perturbation. This is accomplished by writing thickness as the minimum of a
–compact
family of smooth functions in order to apply a theorem of Clarke. We give a number
of applications, including a classification of the “supercoiled helices” formed by
critical curves with no self-contacts (constrained by curvature alone) and an explicit
but surprisingly complicated description of the “clasp” junctions formed when one
rope is pulled tight over another.