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).

Freedman–He–Wang conjecture

The Freedman–He–Wang conjecture (1994) stated that the Möbius energy of nontrivial links in $\Bbb R^3$ is minimized by the stereographic projection of the standard Hopf link. This was proved in 2012 by Ian Agol, Fernando C. Marques and André Neves, by using Almgren–Pitts min-max theory.^[1]

References

1. "[1205.0825] Min-max theory and the energy of links". arxiv.org.

• O'Hara, Jun (1991), "Energy of a knot", Topology 30 (2): 241–247, doi:10.1016/0040-9383(91)90010-2, MR 1098918.
• Freedman, Michael H.; He, Zheng-Xu; Wang, Zhenghan (1994), "Möbius energy of knots and unknots", Annals of Mathematics, Second Series 139 (1): 1–50, doi:10.2307/2946626, MR 1259363.