M\"obius invariance of knot energy

Steve Bryson; Michael H. Freedman; Zheng-Xu He; Zhenghan Wang

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M\"obius invariance of knot energy

Steve Bryson·Michael H. Freedman·Zheng-Xu He·Zhenghan Wang

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Abstract

A physically natural potential energy for simple closed curves in $\bold R^3$ is shown to be invariant under M\"obius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot occur; minimizers within prime knot types exist and are regular. Finally, the number of knot types with energy less than any constant $M$ is estimated.

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Steve Bryson, Michael H. Freedman, Zheng-Xu He, Zhenghan Wang. 1993-01-01. M\"obius invariance of knot energy. https://arxiv.org/abs/math/9301212

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