It was discovered by Jun O'Hara, who demonstrated that the energy blows up as the knot's strands get close to one another.
[1] 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 Michael Freedman, Zheng-Xu He, and Zhenghan Wang (1994) who used it to show the existence of a
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.
[2] Conjecturally, there is no energy minimizer for composite knots.
Robert B. Kusner and John M. 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).
are the ten-dimensional group of angle-preserving diffeomorphisms generated by inversion in 2-spheres.
A knot is created by beginning with a one-dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop.
The idea of knot equivalence is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space.
Algorithms exist to solve this problem, with the first given by Wolfgang Haken in the late 1960s.
[4] Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is.
[5] We shall picture a knot by a smooth curve rather than by a polygon.
The singularities of the planar diagram will be called crossing points and the regions into which it subdivides the plane regions of the diagram.
Apply the Möbius invariance property we complete the proof.
be the arc length parameter of a rectifiable closed curve
It is a short calculation (using the law of cosines) that the first terms transform correctly, i.e., Since
, the regularization term of (1) is the elementary integral Let
The Freedman–He–Wang conjecture (1994) stated that the Möbius energy of nontrivial links in
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.
be a link of 2 components, i.e., a pair of rectifiable closed curves in Euclidean three-space with
If two circles are very far from each other, the cross energy can be made arbitrarily small.
So we are interested in the minimal energy of non-split links.
Note that the definition of the energy extends to any 2-component link in
The Möbius energy has the remarkable property of being invariant under conformal transformations of
This condition is called the conformal invariance property of the Möbius cross energy.
(the standard Hopf link up to orientation and reparameterization).
from the torus to the sphere by The Gauss map of a link
is contained in an oriented affine hyperplane with unit normal vector