Error analysis of the minimum distance energy of a polygonal knot and the m?bius energy of an approximating curve

摘要

Energy minimizing smooth knot configurations have long been approximated by finding knotted polygons that minimize discretized versions of the given energy. However, for most knot energy functionals, the question remains open on whether the minimum polygonal energies are "close" to the minimum smooth energies. In this paper, we determine an explicit bound between the Minimum-Distance Energy of a polygon and the M?bius Energy of a piecewise-C~2 knot inscribed in the polygon. This bound is written in terms of the ropelength and the number of edges and can be used to determine an upper bound for the minimum M?bius Energy for different knot types.