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.[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).
Recall that the Möbius transformations of the 3-sphere
are the ten-dimensional group of angle-preserving diffeomorphisms generated by inversion in 2-spheres. For example, the inversion in the sphere
is defined by
![{\displaystyle \mathbf {x} \to \mathbf {a} +{\rho ^{2} \over |\mathbf {x} -\mathbf {a} |^{2}}\cdot (\mathbf {x} -\mathbf {a} ).}](//wikimedia.org/api/rest_v1/media/math/render/svg/2ed66825b12c76b219859994b237ea7a2605989b)
Consider a rectifiable simple curve
in the Euclidean
3-space
, where
belongs to
or
. Define its energy by
![{\displaystyle E(\gamma )=\iint \left\{{\frac {1}{|\gamma (u)-\gamma (v)|^{2}}}-{\frac {1}{D(\gamma (u),\gamma (v))^{2}}}\right\}|{\dot {\gamma }}(u)||{\dot {\gamma }}(v)|\,du\,dv,}](//wikimedia.org/api/rest_v1/media/math/render/svg/6ff83cc780faf5d235ede1f6bd78813bbd558af1)
where
is the shortest arc
distance between
and
on the curve. The second term of the
integrand is called a
regularization. It is easy to see that
is
independent of parametrization and is unchanged if
is changed by a similarity of
. Moreover, the energy of any line is 0, the energy of any circle is
. In fact, let us use the arc-length parameterization. Denote by
the length of the curve
. Then
![{\displaystyle E(\gamma )=\int _{-\ell /2}^{\ell /2}{}dx\int _{x-\ell /2}^{x+\ell /2}\left[{1 \over |\gamma (x)-\gamma (y)|^{2}}-{1 \over |x-y|^{2}}\right]dy.}](//wikimedia.org/api/rest_v1/media/math/render/svg/5e43d9199c18fbaa3d6a68120731f9ea2827e48c)
Let
denote a unit circle. We have
![{\displaystyle |\gamma _{0}(x)-\gamma _{0}(y)|^{2}={\left(2\sin {\tfrac {1}{2}}(x-y)\right)^{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/86506f062bb339174bacb418115225b8dc27e2ab)
and consequently,
![{\displaystyle {\begin{aligned}E(\gamma _{0})&=\int _{-\pi }^{\pi }{}dx\int _{x-\pi }^{x+\pi }\left[{1 \over \left(2\sin {\tfrac {1}{2}}(x-y)\right)^{2}}-{1 \over |x-y|^{2}}\right]dy\\&=4\pi \int _{0}^{\pi }\left[{1 \over \left(2\sin(y/2)\right)^{2}}-{1 \over |y|^{2}}\right]dy\\&=2\pi \int _{0}^{\pi /2}\left[{1 \over \sin ^{2}y}-{1 \over |y|^{2}}\right]dy\\&=2\pi \left[{1 \over u}-\cot u\right]_{u=0}^{\pi /2}=4\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a6f5bb41c38af406eadd5473e037bb9b040c7c42)
since
.