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Möbius energy of knots and unknots. (English) Zbl 0817.57011

The authors prove:
– the Möbius invariance of the energy of a simple closed curve in \(\mathbb{R}^ 3\) and that the energy bounds the average crossing number of curves in \(\mathbb{R}^ 3\);
– the existence of curves which minimize the energy in the family of loops representing any given irreducible knot;
– curves of finite energy are topologically tame and
– variational formulas for the gradient of the energy.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
58E99 Variational problems in infinite-dimensional spaces
53A04 Curves in Euclidean and related spaces
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