Open Access. Powered by Scholars. Published by Universities.®
Physical Sciences and Mathematics Commons™
Open Access. Powered by Scholars. Published by Universities.®
Articles 1 - 2 of 2
Full-Text Articles in Physical Sciences and Mathematics
Ropelength Criticality, Jason Cantarella, Joseph H.G. Fu, Robert B. Kusner, John M. Sullivan
Ropelength Criticality, Jason Cantarella, Joseph H.G. Fu, Robert B. Kusner, John M. Sullivan
Robert Kusner
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 …
Criticality For The Gehring Link Problem, Jason Cantarella, Joseph H.G. Fu, Robert Kusner, John M. Sullivan, Nancy C. Wrinkle
Criticality For The Gehring Link Problem, Jason Cantarella, Joseph H.G. Fu, Robert Kusner, John M. Sullivan, Nancy C. Wrinkle
Robert Kusner
In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring’s problem to deal with links in a fixed link-homotopy class: we prove ropelength minimizers exist and introduce a theory of ropelength criticality.
Our balance criterion is a set of necessary and sufficient conditions for criticality, based on a strengthened, infinite-dimensional version of the Kuhn–Tucker theorem. We use this to prove that every critical link …