We present new computations of approximately length-minimizing polygons with
fixed thickness. These curves model the centerlines of "tight" knotted tubes
with minimal length and fixed circular cross-section. Our curves approximately
minimize the ropelength (or quotient of length and thickness) for polygons in
their knot types. While previous authors have minimized ropelength for polygons
using simulated annealing, the new idea in our code is to minimize length over
the set of polygons of thickness at least one using a version of constrained
gradient descent.
We rewrite the problem in terms of minimizing the length of the polygon
subject to an infinite family of differentiable constraint functions. We prove
that the polyhedral cone of variations of a polygon of thickness one which do
not decrease thickness to first order is finitely generated, and give an
explicit set of generators. Using this cone we give a first-order minimization
procedure and a Karush-Kuhn-Tucker criterion for polygonal ropelength
criticality.
Our main numerical contribution is a set of 379 almost-critical prime knots
and links, covering all prime knots with no more than 10 crossings and all
prime links with no more than 9 crossings. For links, these are the first
published ropelength figures, and for knots they improve on existing figures.
We give new maps of the self-contacts of these knots and links, and discover
some highly symmetric tight knots with particularly simple looking self-contact
maps.Comment: 45 pages, 16 figures, includes table of data with upper bounds on
ropelength for all prime knots with no more than 10 crossings and all prime
links with no more than 9 crossing
