Möbius energy

In mathematics, the Möbius energy of a knot is a particular knot energy, i.e. a functional on the space of knots. It was discovered by Jun O'Hara, who demonstrated that the energy blows up as the knot's strands get close to one another. This is a useful property because it prevents self-intersection and ensures the result under gradient descent is of the same knot type.

Invariance of Möbius energy under Möbius transformations was demonstrated by Freedman, He, and Wang (1994) who used it to show the existence of a C^1,1 energy minimizer in each isotopy class of a prime knot. They also showed the minimum energy of any knot conformation is achieved by a round circle.

Conjecturally, there is no energy minimizer for composite knots. Kusner and Sullivan have done computer experiments with a discretized version of the Möbius energy and concluded that there should be no energy minimizer for the knot sum of two trefoils (although this is not a proof).

Recall that the Möbius transformations of the 3-sphere ${\displaystyle S^{3}={\mathbf {R}}^{3}\cup \infty }$ are the ten-dimensional group of angle-preserving diffeomorphisms generated by inversion in 2-spheres.For example, the inversion in the sphere ${\displaystyle \{{\mathbf {v}}\in {\mathbf {R}}^{3}\colon |{\mathbf {v}}-{\mathbf {a}}|=\rho \}}$ is defined by ${\displaystyle {\mathbf {x}}\to {\mathbf {a}}+{\rho ^{2} \over |{\mathbf {x}}-{\mathbf {a}}|^{2}}\cdot ({\mathbf {x}}-{\mathbf {a}}).}$

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