Knot Tightening by Constrained Gradient Descent

We present new computations of approximately length-minimizing polygons with fixedthickness. These curves model the centerlines of “tight” knotted tubes with minimal length and fixedcircular cross-section. Our curves approximately minimize the ropelength (or quotient of length andthickness) for polygons in their knot types. While previous authors have minimized ropelength forpolygons using simulated annealing, the new idea in our code is to minimize length over the set ofpolygons 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 infinitefamily of differentiable constraint functions. We prove that the polyhedral cone of variations of apolygon of thickness one which do not decrease thickness to first order is finitely generated, andgive an explicit set of generators. Using this cone we give a first-order minimization procedure anda Karush-Kuhn-Tucker criterion for polygonal ropelength criticality.Our main numerical contribution is a set of 379 almost-critical prime knots and links of 10 or fewer crossings. For links, these are the first published ropelength figures, and for knots they improveon existing figures. We give new maps of the self-contacts of these knots and links, and discoversome highly symmetric tight knots with particularly simple looking self-contact maps.