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 Michael Freedman, Zheng-Xu He, and Zhenghan Wang (1994) who used it to show the existence of a ${\displaystyle C^{1,1}}$ 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. 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 ${\displaystyle S^{3}=\mathbf {R} ^{3}\cup \infty }$ 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 ${\displaystyle \mathbf {x} \to \mathbf {a} +{\rho ^{2} \over |\mathbf {x} -\mathbf {a} |^{2}}\cdot (\mathbf {x} -\mathbf {a} ).}$

Consider a rectifiable simple curve ${\displaystyle \gamma (u)}$ in the Euclidean 3-space ${\displaystyle \mathbf {R} ^{3}}$, where ${\displaystyle u}$ belongs to ${\displaystyle \mathbf {R} ^{1}}$ or ${\displaystyle S^{1}}$. Define its energy by

${\displaystyle E(\gamma )=\iint \left\{{\frac {1}{|\gamma (u)-\gamma (v)|^{2}}}-{\frac {1}{D(\gamma (u),\gamma (v))^{2}}}\right\}|{\dot {\gamma }}(u)||{\dot {\gamma }}(v)|\,du\,dv,}$

where ${\displaystyle D(\gamma (u),\gamma (v))}$ is the shortest arc distance between ${\displaystyle \gamma (u)}$ and ${\displaystyle \gamma (v)}$ on the curve. The second term of the integrand is called a regularization. It is easy to see that ${\displaystyle E(\gamma )}$ is independent of parametrization and is unchanged if ${\displaystyle \gamma }$ is changed by a similarity of ${\displaystyle \mathbf {R} ^{3}}$. Moreover, the energy of any line is 0, the energy of any circle is ${\displaystyle 4}$. In fact, let us use the arc-length parameterization. Denote by ${\displaystyle \ell }$ the length of the curve ${\displaystyle \gamma }$. Then

${\displaystyle E(\gamma )=\int _{-\ell /2}^{\ell /2}{}dx\int _{x-\ell /2}^{x+\ell /2}\left[{1 \over |\gamma (x)-\gamma (y)|^{2}}-{1 \over |x-y|^{2}}\right]dy.}$

Let ${\displaystyle \gamma _{0}(t)=(\cos t,\sin t,0)}$ denote a unit circle. We have

${\displaystyle |\gamma _{0}(x)-\gamma _{0}(y)|^{2}={\left(2\sin {\tfrac {1}{2}}(x-y)\right)^{2}}}$

and consequently,

${\displaystyle {\begin{aligned}E(\gamma _{0})&=\int _{-\pi }^{\pi }{}dx\int _{x-\pi }^{x+\pi }\left[{1 \over \left(2\sin {\tfrac {1}{2}}(x-y)\right)^{2}}-{1 \over |x-y|^{2}}\right]dy\\&=4\pi \int _{0}^{\pi }\left[{1 \over \left(2\sin(y/2)\right)^{2}}-{1 \over |y|^{2}}\right]dy\\&=2\pi \int _{0}^{\pi /2}\left[{1 \over \sin ^{2}y}-{1 \over |y|^{2}}\right]dy\\&=2\pi \left[{1 \over u}-\cot u\right]_{u=0}^{\pi /2}=4\end{aligned}}}$

since ${\displaystyle {\frac {1}{u}}-\cot u={\frac {u}{3}}-\cdots }$.