Polygonal knot space near ropelength-minimized knots
Equilateral knots, polygonal knots, perturbation, thick knots, satellite knots, local knots, ropelength
For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r < R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which one can obtain small trefoil and figure-eight summands and use Monte Carlo simulations to estimate the relative probabilities of these structures as a function of the number of edges.
Journal of Knot Theory and Its Ramifications