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For a polygonal knot K, it is shown that a tube of radius R(K), the polygonalthickness radius, is an embedded torus. Given a thick configuration K,perturbations of size r<R(K) define satellite structures, or local knotting. Weexplore knotting within these tubes both theoretically and numerically. Weprovide bounds on perturbation radii for which we can see small trefoil andfigure-eight summands and use Monte Carlo simulations to approximate therelative probabilities of these structures as a function of the number ofedges.

POLYGONAL KNOT SPACE NEAR ROPELENGTH-MINIMIZED KNOTS