Möbius energy

On the left, the unknot, and a knot equivalent to it. It can be more difficult to determine whether complex knots, such as the one on the right, are equivalent to the unknot.

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.[3] Mathematically, we can say a knot ${\displaystyle K}$ is an injective and continuous function ${\displaystyle K\colon [0,1]\to \mathbb {R} ^{3}}$ with ${\displaystyle K(0)=K(1)}$. Topologists consider knots and other entanglements such as links and braids to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot. 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. A mathematical definition is that two knots ${\displaystyle K_{1},K_{2}}$ are equivalent if there is an orientation-preserving homeomorphism ${\displaystyle h\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}$ with ${\displaystyle h(K_{1})=K_{2}}$, and this is known to be equivalent to existence of ambient isotopy.

The basic problem of knot theory, the recognition problem, is determining the equivalence of two knots. 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.[4] The special case of recognizing the unknot, called the unknotting problem, is of particular interest.[5] We shall picture a knot by a smooth curve rather than by a polygon. A knot will be represented by a planar diagram. The singularities of the planar diagram will be called crossing points and the regions into which it subdivides the plane regions of the diagram. At each crossing point, two of the four corners will be dotted to indicate which branch through the crossing point is to be thought of as one passing under the other. We number any one region at random, but shall fix the numbers of all remaining regions such that whenever we cross the curve from right to left we must pass from region number ${\displaystyle k}$ to the region number ${\displaystyle k+1}$. Clearly, at any crossing point ${\displaystyle c}$, there are two opposite corners of the same number ${\displaystyle k}$ and two opposite corners of the numbers ${\displaystyle k-1}$ and ${\displaystyle k+1}$, respectively. The number ${\displaystyle k}$ is referred as the index of ${\displaystyle c}$. The crossing points are distinguished by two types: the right handed and the left handed, according to which branch through the point passes under or behind the other. At any crossing point of index ${\displaystyle k}$ two dotted corners are of numbers ${\displaystyle k}$ and ${\displaystyle k+1}$, respectively, two undotted ones of numbers ${\displaystyle k-1}$ and ${\displaystyle k+1}$. The index of any corner of any region of index ${\displaystyle k}$ is one element of ${\displaystyle \{k\pm 1,k\}}$. We wish to distinguish one type of knot from another by knot invariants. There is one invariant which is quite simple. It is Alexander polynomial ${\displaystyle \Delta _{K}(t)}$ with integer coefficient. The Alexander polynomial is symmetric with degree ${\displaystyle n}$: ${\displaystyle \Delta _{K}(t^{-1})t^{n-1}=\Delta _{K}(t)}$ for all knots ${\displaystyle K}$ of ${\displaystyle n>0}$ crossing points. For example, the invariant ${\displaystyle \Delta _{K}(t)}$ of an unknotted curve is 1, of an trefoil knot is ${\displaystyle t^{2}-t+1}$.

• 
  The left handed trefoil knot.
• 
  The right handed trefoil knot.

Let

${\displaystyle \omega ({\boldsymbol {x}})={\frac {1}{4\pi }}\varepsilon _{ijk}{x^{i}dx^{j}\wedge dx^{k} \over |{\boldsymbol {x}}|^{3}}}$ denote the standard surface element of ${\displaystyle S^{2}}$.

We have

${\displaystyle \mathrm {link} (\gamma _{1},\gamma _{2})=\int _{{\boldsymbol {x}}\in \gamma _{1},{\boldsymbol {y}}\in \gamma _{2}}\omega ({\boldsymbol {x}}-{\boldsymbol {y}})}$

${\displaystyle \int _{S^{2}}\omega ({\boldsymbol {x}})={\frac {1}{4\pi }}\int _{S^{2}}\varepsilon _{ijk}x^{i}dx^{j}dx^{k}=1,\qquad \omega (\lambda {\boldsymbol {x}})=\omega ({\boldsymbol {x}}){\rm {sign}}\lambda ,\quad {\rm {for}}\quad \lambda \in \mathbb {R} ^{*}.}$

For the knot ${\displaystyle \gamma$ :[0,1]\rightarrow \mathbb {R} ^{3}} , ${\displaystyle \gamma (0)=\gamma (1)}$,

${\displaystyle \int _{t_{1}<t_{2}<t_{3}<t_{4}<1}\omega (\gamma (t_{1})-\gamma (t_{3}))\wedge \omega (\gamma (t_{2})-\gamma (t_{4}))}$

${\displaystyle +\int _{t_{1}<t_{2}<t_{3},{\boldsymbol {x}}\in \mathbb {R} ^{3}\setminus \gamma ([0,1])}\omega (\gamma (t_{1})-{\boldsymbol {x}})\wedge \omega (\gamma (t_{2})-{\boldsymbol {x}})\wedge \omega (\gamma (t_{3})-{\boldsymbol {x}})}$

does not change, if we change the knot ${\displaystyle \gamma }$ in its equivalence class.

Let ${\displaystyle \gamma }$ be a closed curve in ${\displaystyle \mathbb {R} ^{3}}$ and ${\displaystyle T}$ a Möbius transformation of ${\displaystyle S^{3}=\mathbb {R} ^{3}\cup \infty }$. If ${\displaystyle T(\gamma )}$ is contained in ${\displaystyle \mathbb {R} ^{3}}$ then ${\displaystyle E(T(\gamma ))=E(\gamma )}$. If ${\displaystyle T(\gamma )}$ passes through ${\displaystyle \infty }$ then ${\displaystyle E(T(\gamma ))=E(\gamma )-4}$.

Theorem A. Among all rectifiable loops ${\displaystyle \gamma \colon S^{1}\to \mathbb {R} ^{3}}$, round circles have the least energy ${\displaystyle E({\text{round circle}})=4}$ and any ${\displaystyle \gamma }$ of least energy parameterizes a round circle.

Proof of Theorem A. Let ${\displaystyle T}$ be a Möbius transformation sending a point of ${\displaystyle \gamma }$ to infinity. The energy ${\displaystyle E(T(\gamma ))\geq 0}$ with equality holding if and only if ${\displaystyle T(\gamma )}$ is a straight line. Apply the Möbius invariance property we complete the proof.

Proof of Möbius Invariance Property. It is sufficient to consider how ${\displaystyle I}$, an inversion in a sphere, transforms energy. Let ${\displaystyle u}$ be the arc length parameter of a rectifiable closed curve ${\displaystyle \gamma }$, ${\displaystyle u\in \mathbb {R} /\ell \mathbb {Z} }$. Let

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

(1)

and

${\displaystyle {\begin{aligned}E_{\varepsilon }(I\circ \gamma )=&\iint _{|u-v|\geq \varepsilon }\left({\frac {1}{|I\circ \gamma (u)-I\circ \gamma (v)|^{2}}}-{\frac {1}{(D(I\circ \gamma (u),I\circ \gamma (v)))^{2}}}\right)\\&\qquad \times \|I'(\gamma (u))\|\cdot \|I'(\gamma (v))\|\,du\,dv.\end{aligned}}}$

(2)

Clearly, ${\displaystyle E(\gamma )=\lim _{\varepsilon \to 0}E_{\varepsilon }(\gamma )}$ and ${\displaystyle E(I\circ \gamma )=\lim _{\varepsilon \to 0}E_{\varepsilon }(I\circ \gamma )}$. It is a short calculation (using the law of cosines) that the first terms transform correctly, i.e.,

${\displaystyle {\frac {\|I'(\gamma (u))\|\cdot \|I'(\gamma (v))\|}{|I(\gamma (u))-I(\gamma (v))|^{2}}}={\frac {1}{|\gamma (u)-\gamma (v)|^{2}}}.}$

Since ${\displaystyle u}$ is arclength for ${\displaystyle \gamma }$, the regularization term of (1) is the elementary integral

${\displaystyle \int _{u=0}^{\ell }\left[2\int _{v=\varepsilon }^{\ell /2}{\frac {1}{v^{2}}}\,dv\right]\,du=4-{\frac {2\ell }{\varepsilon }}.}$

(3)

Let ${\displaystyle s}$ be an arclength parameter for ${\displaystyle I\circ \gamma }$. Then ${\displaystyle ds(u)/du=\|I'(\gamma (u))\|}$ where ${\displaystyle \|I'(\gamma (u))\|=f(u)}$ denotes the linear expansion factor of ${\displaystyle I'}$. Since ${\displaystyle \gamma (u)}$ is a Lipschitz function and ${\displaystyle I'}$ is smooth, ${\displaystyle f(u)}$ is Lipschitz, hence, it has weak derivative ${\displaystyle f'(u)\in L^{\infty }}$.

${\displaystyle {\begin{aligned}{\rm {{regularization}(2)=}}&\int _{u\in \mathbf {R} /\ell \mathbf {Z} }\left[\int _{|v-u|\geq \varepsilon }{\frac {|(I\circ \gamma )'(v)|\,dv}{D(I\circ \gamma (u),I\circ \gamma (v))^{2}}}\right]|(I\circ \gamma )'(u)|\,du\\=&\int _{\mathbf {R} /\ell \mathbf {Z} }\left[{\frac {4}{L}}-{\frac {1}{\varepsilon _{+}}}-{\frac {1}{\varepsilon _{-}}}\right]\,ds,\end{aligned}}}$

(4)

where ${\displaystyle L={\rm {{Length}(I(\gamma ))}}}$ and

${\displaystyle {\begin{aligned}\varepsilon _{+}&=\varepsilon _{+}(u)=D((I\circ \gamma )(u),(I\circ \gamma )(u+\varepsilon ))=s(u+\varepsilon )-s(u)\\&=\int _{u}^{u+\varepsilon }f(w)\,dw=f(u)\varepsilon +\varepsilon ^{2}\int _{0}^{1}(1-t)f'(u+\varepsilon t)\,dt\end{aligned}}}$

and

${\displaystyle \varepsilon _{-}=\varepsilon _{-}(u)=D((I\circ \gamma )(u-\varepsilon ),(I\circ \gamma )(u))=f(u)\varepsilon -\varepsilon ^{2}\int _{0}^{1}(1-t)f'(u-\varepsilon t)\,dt.}$

Since ${\displaystyle |f'(w)|}$ is uniformly bounded, we have

${\displaystyle {\begin{aligned}{\frac {1}{\varepsilon _{+}}}=&{\frac {1}{f(u)\varepsilon }}\left[{1+{\varepsilon \over f(u)}\int _{0}^{1}(1-t)f'(u+\varepsilon t)\,dt}\right]^{-1}\\=&{\frac {1}{f(u)\varepsilon }}\left[1-{\frac {\varepsilon }{f(u)}}\int _{0}^{1}(1-t)f'(u+\varepsilon t)\,dt+O(\varepsilon ^{2})\right]\\=&{\frac {1}{f(u)\varepsilon }}-{\frac {1}{f(u)^{2}}}\int _{0}^{1}(1-t)f'(u+\varepsilon t)\,dt+O(\varepsilon ).\end{aligned}}}$

Similarly, ${\displaystyle {\frac {1}{\varepsilon _{-}}}={\frac {1}{f(u)\varepsilon }}+{\frac {1}{f(u)^{2}}}\int _{0}^{1}(1-t)f'(u-\varepsilon t)\,dt+O(\varepsilon ).}$

Then by (4)

${\displaystyle {\begin{aligned}{\rm {{regularization}\ (2)=}}&4-\int _{0}^{\ell }{\frac {2}{\varepsilon }}\,du+O(\varepsilon )\\&+\int _{u=0}^{\ell }\int _{t=0}^{1}{\frac {(1-t)}{f(u)}}[f'(u+\varepsilon t)-f'(u-\varepsilon t)]\,du\,dt\\=&4-{\frac {2\ell }{\varepsilon }}+O(\varepsilon ).\end{aligned}}}$

(5)

Comparing (3) and (5), we get ${\displaystyle E_{\varepsilon }(\gamma )-E_{\varepsilon }(I\circ \gamma )=O(\varepsilon );}$ hence, ${\displaystyle E(\gamma )=E(I\circ \gamma )}$.

For the second assertion, let ${\displaystyle I}$ send a point of ${\displaystyle \gamma }$ to infinity. In this case ${\displaystyle L=\infty }$ and, thus, the constant term 4 in (5) disappears.

The Freedman–He–Wang conjecture (1994) stated that the Möbius energy of nontrivial links in ${\displaystyle \mathbb {R} ^{3}}$ 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.[6] Let ${\displaystyle \gamma _{i}:S^{1}\rightarrow \mathbb {R} ^{3}}$, ${\displaystyle i=1,2,}$ be a link of 2 components, i.e., a pair of rectifiable closed curves in Euclidean three-space with ${\displaystyle \gamma _{1}(S^{1})\cap \gamma _{2}(S^{1})=\emptyset }$. The Möbius cross energy of the link ${\displaystyle (\gamma _{1},\gamma _{2})}$ is defined to be

${\displaystyle E(\gamma _{1},\gamma _{2})=\int _{S^{1}\times S^{1}}{\frac {|{\dot {\gamma }}_{1}(s)||{\dot {\gamma }}_{2}(t)|}{|\gamma _{1}(s)-\gamma _{2}(t)|^{2}}}\,ds\,dt.}$

The linking number of ${\displaystyle (\gamma _{1},\gamma _{2})}$ is defined by letting

${\displaystyle {\begin{aligned}\mathrm {link} (\gamma _{1},\gamma _{2})&=\,{\frac {1}{4\pi }}\oint _{\gamma _{1}}\oint _{\gamma _{2}}{\frac {\mathbf {r} _{1}-\mathbf {r} _{2}}{|\mathbf {r} _{1}-\mathbf {r} _{2}|^{3}}}\cdot (d\mathbf {r} _{1}\times d\mathbf {r} _{2})\\&={\frac {1}{4\pi }}\int _{S^{1}\times S^{1}}{\frac {\mathrm {det} ({\dot {\gamma }}_{1}(s),{\dot {\gamma }}_{2}(t),\gamma _{1}(s)-\gamma _{2}(t))}{|\gamma _{1}(s)-\gamma _{2}(t)|^{3}}}\,ds\,dt.\end{aligned}}}$

${\displaystyle \cdots }$
	linking number −2	linking number −1	linking number 0
					${\displaystyle \cdots }$
		linking number 1	linking number 2	linking number 3

It is not difficult to check that ${\displaystyle E(\gamma _{1},\gamma _{2})\geq 4\pi |{\rm {link}}(\gamma _{1},\gamma _{2})|}$. If two circles are very far from each other, the cross energy can be made arbitrarily small. If the linking number ${\displaystyle \mathrm {link} (\gamma _{1},\gamma _{2})}$ is non-zero, the link is called non-split and for the non-split link, ${\displaystyle E(\gamma _{1},\gamma _{2})\geq 4\pi }$. 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 ${\displaystyle \mathbb {R} ^{n}}$. The Möbius energy has the remarkable property of being invariant under conformal transformations of ${\displaystyle \mathbb {R} ^{3}}$. This property is explained as follows. Let ${\displaystyle F:\mathbb {R} ^{3}\rightarrow {S^{3}}}$ denote a conformal map. Then ${\displaystyle E(\gamma _{1},\gamma _{2})=E(F\circ \gamma _{1},F\circ \gamma _{2}).}$ This condition is called the conformal invariance property of the Möbius cross energy.

Main Theorem. Let ${\displaystyle \gamma _{i}:S^{1}\rightarrow \mathbb {R} ^{3}}$, ${\displaystyle i=1,2,}$ be a non-split link of 2 components link. Then ${\displaystyle E(\gamma _{1},\gamma _{2})\geq 2\pi ^{2}}$. Moreover, if ${\displaystyle E(\gamma _{1},\gamma _{2})=2\pi ^{2}}$ then there exists a conformal map ${\displaystyle F:\mathbb {R} ^{3}\rightarrow {S^{3}}}$ such that ${\displaystyle F\circ \gamma _{1}(t)=(\cos t,\sin t,0,0)}$ and ${\displaystyle F\circ \gamma _{2}(t)=(0,0,\cos t,\sin t)}$ (the standard Hopf link up to orientation and reparameterization).

Given two non-intersecting differentiable curves ${\displaystyle \gamma _{1},\gamma _{2}\colon S^{1}\rightarrow \mathbb {R} ^{3}}$, define the Gauss map ${\displaystyle \Gamma }$ from the torus to the sphere by

${\displaystyle \Gamma (s,t)={\frac {\gamma _{1}(s)-\gamma _{2}(t)}{|\gamma _{1}(s)-\gamma _{2}(t)|}}.}$

The Gauss map of a link ${\displaystyle (\gamma _{1},\gamma _{2})}$ in ${\displaystyle \mathbf {R} ^{4}}$, denoted by ${\displaystyle g=G(\gamma _{1},\gamma _{2})}$, is the Lipschitz map ${\displaystyle g:S^{1}\times S^{1}\to S^{3}}$ defined by ${\displaystyle g(s,t)={\frac {\gamma _{1}(s)-\gamma _{2}(t)}{|\gamma _{1}(s)-\gamma _{2}(t)|}}.}$ We denote an open ball in ${\displaystyle \mathbf {R} ^{4}}$, centered at ${\displaystyle \mathbf {x} }$ with radius ${\displaystyle r}$, by ${\displaystyle B_{r}^{4}(\mathbf {x} )}$. The boundary of this ball is denoted by ${\displaystyle S_{r}^{3}(\mathbf {x} )}$. An intrinsic open ball of ${\displaystyle S^{3}}$, centered at ${\displaystyle \mathbf {p} \in S^{3}}$ with radius ${\displaystyle r}$, is denoted by ${\displaystyle B_{r}(\mathbf {p} )}$. We have

${\displaystyle {\frac {\partial g}{\partial s}}={{\dot {\gamma }}_{1}-\langle g,{\dot {\gamma }}_{1}\rangle g \over |\gamma _{1}-\gamma _{2}|}\quad {\mbox{and}}\quad {\frac {\partial g}{\partial t}}=-{{\dot {\gamma }}_{2}-\langle g,{\dot {\gamma }}_{2}\rangle g \over |\gamma _{1}-\gamma _{2}|}.}$

Thus,

${\displaystyle {\begin{aligned}\left|{\frac {\partial g}{\partial s}}\right|^{2}\left|{\frac {\partial g}{\partial t}}\right|^{2}-\left\langle {\frac {\partial g}{\partial s}},{\frac {\partial g}{\partial t}}\right\rangle ^{2}&\leq \left|{\frac {\partial g}{\partial s}}\right|^{2}\left|{\frac {\partial g}{\partial t}}\right|^{2}\\&={\frac {|{\dot {\gamma }}_{1}|^{2}-\langle g,{\dot {\gamma }}_{1}\rangle ^{2}}{|\gamma _{1}-\gamma _{2}|^{2}}}{\frac {|{\dot {\gamma }}_{2}|^{2}-\langle g,{\dot {\gamma }}_{2}\rangle ^{2}}{|\gamma _{1}-\gamma _{2}|^{2}}}\\&\leq {\frac {|{\dot {\gamma }}_{1}|^{2}|{\dot {\gamma }}_{2}|^{2}}{|\gamma _{1}-\gamma _{2}|^{4}}}.\end{aligned}}}$

It follows that for almost every ${\displaystyle (s,t)\in S^{1}\times S^{1}}$, ${\displaystyle |{\rm {Jac\,}}g|(s,t)\leq {\frac {|{\dot {\gamma }}_{1}(s)||{\dot {\gamma }}_{2}(t)|}{|\gamma _{1}(s)-\gamma _{2}(t)|^{2}}}.}$ If equality holds at ${\displaystyle (s,t)}$, then ${\displaystyle \langle {\dot {\gamma }}_{1}(s),{\dot {\gamma }}_{2}(t)\rangle =\langle {\dot {\gamma }}_{1}(s),\gamma _{1}(s)-\gamma _{2}(t)\rangle =\langle {\dot {\gamma }}_{2}(t),\gamma _{1}(s)-\gamma _{2}(t)\rangle =0.}$

${\displaystyle {\mathbf {M} }(C)\leq \int _{S^{1}\times S^{1}}|{\rm {Jac\,}}g|\,ds\,dt\leq E(\gamma _{1},\gamma _{2}).}$

If the link ${\displaystyle (\gamma _{1},\gamma _{2})}$ is contained in an oriented affine hyperplane with unit normal vector ${\displaystyle \mathbf {p} \in S^{3}}$ compatible with the orientation, then ${\displaystyle C={\rm {link}}(\gamma _{1},\gamma _{2})\cdot \partial B_{\pi /2}(-\mathbf {p} ).}$

• Adams, Colin (2004). The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society. ISBN 9780821836781.
• Hass, Joel (April–May 1998). "Algorithms for recognizing knots and 3-manifolds". Chaos, Solitons and Fractals. 9 (4–5): 569–581. arXiv:math/9712269. Bibcode:1998CSF.....9..569H. doi:10.1016/S0960-0779(97)00109-4. S2CID 7381505.
• Sossinsky, Alexei (2002). Knots, mathematics with a twist. Harvard University Press. ISBN 9780674009448.

Footnotes