While reading the paper, A simple energy function for knots, I understand that the authors have proved the two conditions of the first page for the normal projection energy of a knot. But I failed to find the proofs that the energy is minimal for the canonical form and invariant during evolution. It would be a great favor if anyone could give me hint about how to do or where to find the proofs. Thanks in advance!
2026-03-26 11:18:25.1774523905
Minimizer and invariance of normal projection energy of a knot
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You shouldn't expect a knot energy functional to be invariant under anything. In good cases you can find a minimal energy representative, but there may be no unique minimal representative. Also, what do you mean by "canonical form?" Knots can normally be represented in multiple ways and even the diagram with the lowest number of crossings is not in general unique.