  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 $C^1$-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.
