Möbius energy of a closed curve in R³ (O’Hara 1991):
E = ∬ ( 1 / |x − y|² − 1 / d(x, y)² ) ds dt,
with d the distance along the curve. E is invariant under Möbius transformations; E ≥ 4, with equality only for round circles; and E ≥ 2πc + 4 for a knot with crossing number c (Freedman, He, Wang 1994). Then E < 6π + 4 implies unknotted.
It is conjectured that E has a unique critical point in the space of embeddings of S¹ ambient isotopic to the round unknot. If true, gradient descent on E unknots any unknot.
The curve is a closed polygon of fixed length. Each step follows the H3/2 Sobolev gradient of E (Yu, Schumacher, Crane 2021) and steps are rejected if they would cause crossings. Minimisers are determined only up to Möbius transformations, so the representative of least bending energy is shown. Vertices are added where strands are close or curvature is high, and removed as the curve relaxes.
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