View source for Möbius energy

{{Use shortened footnotes|date=April 2021}}
In [[mathematics]], the '''Möbius energy''' of a [[knot (mathematics)|knot]] is a particular [[knot energy]], i.e., a [[Functional (mathematics)|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.{{r|OHara1991}}  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 transformation#Higher dimensions|Möbius transformation]]s was demonstrated by [[Michael Freedman]], Zheng-Xu He, and [[Zhenghan Wang]] (1994) who used it to show the existence of a <math>C^{1,1}</math> 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.{{r|FreedmanHeWang1994}}

Conjecturally, there is no energy minimizer for composite knots. [[Robert B. Kusner]] and [[John M. Sullivan (mathematician)|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 
<math>S^3=\mathbf R^3\cup\infty</math> are the ten-dimensional group of angle-preserving diffeomorphisms generated by inversion in 2-spheres. For example, the inversion in the sphere <math>\{\mathbf v\in\mathbf R^3\colon |\mathbf v-\mathbf a|=\rho\}</math> is defined by 
<math>\mathbf x\to\mathbf a+{\rho^2\over |\mathbf x-\mathbf a|^2}\cdot(\mathbf x-\mathbf a).</math>

Consider a rectifiable simple curve <math>\gamma(u)</math> in the Euclidean 
3-space <math>\mathbf R^3</math>, where <math>u</math> belongs to <math>\mathbf R^1</math> or <math>S^1</math>. Define its energy by

:<math>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,</math>

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

:<math>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.
 </math>

Let <math>\gamma_0(t)=(\cos t, \sin t, 0)</math> denote a unit circle. We have

:<math>
 |\gamma_0(x)-\gamma_0(y)|^2={\left(2\sin\tfrac12(x-y)\right)^2}</math>

and consequently,
 
:<math>\begin{align} E(\gamma_0)&=\int_{-\pi}^{\pi}{}dx\int_{x-\pi}^{x+\pi}\left[{1\over \left(2\sin\tfrac12(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^2y}-{1\over|y|^2}\right]dy\\
    &=2\pi\left[{1\over u}-\cot u\right]_{u=0}^{\pi/2}=4
   \end{align}</math>

since <math>\frac1u-\cot u=\frac u3-\cdots</math>.

==Knot invariant<!--[[Knot invariant]] redirects directly here.-->==
{{multiple image
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| image2 = Ochiai unknot.svg
| width2 = 162
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| caption2 = 
| footer = 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-[[dimension]]al line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop.<ref>{{Harvnb|Adams|2004}}; {{Harvnb|Sossinsky|2002}}.</ref>  Mathematically, we can say a knot <math>K</math> is an [[injective]] and [[continuous function]] <math>K\colon[0,1]\to \mathbb{R}^3</math> with <math>K(0)=K(1)</math>. Topologists consider knots and other entanglements such as [[link (knot theory)|links]] and [[Braid theory|braid]]s 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 <math>K_1,K_2</math> are equivalent if there is an [[orientation-preserving]] [[homeomorphism]] <math>h\colon\R^3\to\R^3</math> with <math> h(K_1)=K_2</math>, 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.  [[Algorithm]]s exist to solve this problem, with the first given by [[Wolfgang Haken]] in the late 1960s.{{sfn|Hass|1998}} Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is.{{sfn|Hass|1998}} The special case of recognizing the [[unknot]], called the [[unknotting problem]], is of particular interest.{{r|Hoste2005}}
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 <math>k</math> to the region number <math>k+1</math>. Clearly, at any crossing point <math>c</math>, there are two opposite corners of the same number <math>k</math> and two opposite corners of the  numbers <math>k-1</math> and <math>k+1</math>, respectively. The number <math>k</math> is referred as the index of <math>c</math>. 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 <math>k</math> two dotted corners are of numbers <math>k</math> and <math>k+1</math>, respectively, two undotted ones of  numbers <math>k-1</math> and <math>k+1</math>. The index of any  corner of any region of index <math>k</math> is one element of <math>\{k\pm1,k\}</math>. 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]] <math>\Delta_K(t)</math> with integer coefficient. The Alexander polynomial is symmetric with degree <math>n</math>: <math>\Delta_K(t^{-1})t^{n-1} = \Delta_K(t)</math> for all knots <math>K</math> of  <math>n>0</math> crossing points. For example, the invariant <math>\Delta_K(t)</math> of an unknotted curve is 1, of an trefoil knot is <math>t^2-t+1</math>.

<gallery widths="80px" heights="80px" align="right">
Image:Trefoil knot left.svg|The left handed trefoil knot.
Image:TrefoilKnot_01.svg|The right handed trefoil knot.
</gallery>

Let
:<math>\omega(\boldsymbol{x})=\frac1{4\pi}\varepsilon_{ijk}{x^idx^j\wedge dx^k\over|\boldsymbol{x}|^3}</math> denote the standard  surface element of <math>S^2</math>. 
We have

:<math>\mathrm {link}(\gamma_1,\gamma_2)=\int_{\boldsymbol{x}\in\gamma_1,\boldsymbol{y}\in\gamma_2}\omega(\boldsymbol{x}-\boldsymbol{y})
</math>

:<math>\int_{S^2}\omega(\boldsymbol{x})=\frac1{4\pi}\int_{S^2}\varepsilon_{ijk}x^idx^j dx^k=1,\qquad \omega(\lambda\boldsymbol{x})= \omega(\boldsymbol{x}){\rm {sign}}\lambda, \quad{\rm { for}  }\quad\lambda \in\R^*.
</math>

For the knot <math>\gamma: [0,1] \rightarrow \mathbb R^3</math>,  <math>\gamma(0)=\gamma(1)
</math>,

:<math>\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))</math>

:<math>+\int_{t_1<t_2<t_3,\boldsymbol{x}\in\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})
</math>

does not change, if we change the knot <math>\gamma </math> in its equivalence class.

==Möbius Invariance Property==  
Let <math>\gamma</math> be a closed curve in <math>\R^3</math> and <math>T</math> a Möbius transformation of <math>S^3 = \R^3 \cup \infty</math>. If <math>T(\gamma)</math> is contained in <math>\R^3</math> then <math>E(T(\gamma))=E(\gamma)</math>. If <math>T(\gamma)</math> passes through <math>\infty</math> then <math>E(T(\gamma)) = E(\gamma)-4</math>.

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

'''Proof of Theorem A.''' Let <math>T</math>  be a Möbius transformation sending a point of <math>\gamma</math> to infinity. The energy <math>E(T(\gamma))\ge0</math> with equality holding if and only if <math>T(\gamma)</math> is a straight line. Apply the Möbius invariance property we complete the proof.

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

{{NumBlk|:|<math>E_\varepsilon(\gamma)=\iint_{|u-v|\ge\varepsilon}\left(\frac{1}{|\gamma(u)-\gamma(v)|^2} - \frac{1}{D(\gamma(u),\gamma(v))^2}\right)\,du\,dv</math>|{{EquationRef|1}}}}

and
{{NumBlk|:|<math>\begin{align}
E_\varepsilon(I\circ\gamma)=&\iint_{|u-v|\ge\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{align}</math>|{{EquationRef|2}}}}

Clearly, <math>E(\gamma)=\lim_{\varepsilon\to0}E_\varepsilon(\gamma)</math>  and <math>E(I\circ\gamma)=\lim_{\varepsilon\to0}E_\varepsilon(I\circ\gamma)</math>. It is a short calculation (using the law of cosines) that the first terms transform correctly, i.e.,

:<math>\frac{\|I'(\gamma(u))\|\cdot\|I'(\gamma(v))\|}{|I(
\gamma(u))-I(\gamma(v))|^2}=
\frac{1}{|\gamma(u)-\gamma(v)|^2}.</math>

Since  <math>u</math> is arclength for <math>\gamma</math>, the regularization term of ({{EquationNote|1}}) is the elementary integral

{{NumBlk|:|<math>\int_{u=0}^\ell\left[2\int_{v=\varepsilon}^{\ell/2}\frac{1}{v^2}\,dv
\right]\,du=4-
\frac{2\ell}{\varepsilon}.</math>|{{EquationRef|3}}}}

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

{{NumBlk|:|<math>\begin{align}
\rm{regularization } (2)=&\int_{u\in\mathbf R/\ell\mathbf
Z}\left[\int_{|v-u|\ge\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{align}</math>|{{EquationRef|4}}}}

where <math>L=\rm{Length}(I(\gamma))</math> and

:<math>\begin{align}
\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{align}</math>

and

:<math>
\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.</math>

Since <math>|f'(w)|</math> is uniformly bounded, we have

:<math>\begin{align}
\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{align}</math>

Similarly,
<math>\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).</math>

Then by ({{EquationNote|4}})

{{NumBlk|:|<math>\begin{align}
\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{align}</math>|{{EquationRef|5}}}}

Comparing ({{EquationNote|3}}) and ({{EquationNote|5}}), we get
<math>E_\varepsilon(\gamma)-E_\varepsilon(I\circ\gamma)= O(\varepsilon);</math>
hence, <math>E(\gamma)=E(I\circ\gamma)</math>.

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

==Freedman–He–Wang conjecture==
The '''Freedman–He–Wang conjecture''' (1994) stated that the Möbius energy of nontrivial [[Link (knot theory)|links]] in <math>\mathbb R^3</math> is minimized by the [[stereographic projection]] of the standard [[Hopf link]]. This was proved in 2012 by [[Ian Agol]], [[Fernando Codá Marques|Fernando C. Marques]] and [[André Neves]], by using [[Almgren–Pitts min-max theory]].{{r|AgolMarNeves2012}} Let <math>\gamma_i: S^1 \rightarrow \mathbb R^3</math>, <math>i=1,2,</math>  be a  link  of 2 components, i.e., a pair of rectifiable closed curves in Euclidean three-space  with <math>\gamma_1(S^1) \cap \gamma_2(S^1) = \emptyset</math>. The  Möbius cross energy of the link <math>(\gamma_1,\gamma_2)</math> is defined to be

:<math>
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.
</math>

The linking number of <math>
(\gamma_1,\gamma_2)</math> is defined by letting

:<math>\begin{align}
\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{align}
</math>
{| border=0 cellpadding=5 align="center"
|-valign="center"
|<math>\cdots</math>
|align="center"|[[Image:Linking Number -2.svg|140px]]
|align="center"|[[Image:Linking Number -1.svg|140px]]
|align="center"|[[Image:Linking Number 0.svg|140px]]
|
|
|-valign="center"
|
|align="center"|linking number −2
|align="center"|linking number −1
|align="center"|linking number 0
|
|
|-valign="center"
|
|
|align="center"|[[Image:Linking Number 1.svg|140px]]
|align="center"|[[Image:Linking Number 2.svg|140px]]
|align="center"|[[Image:Linking Number 3.svg|140px]]
|<math>\cdots</math>
|-valign="center"
|
|
|align="center"|linking number 1
|align="center"|linking number 2
|align="center"|linking number 3
|
|}
It is not difficult to check that <math>E(\gamma_1,\gamma_2)\geq 4\pi |{\rm link}(\gamma_1,\gamma_2)|</math>. If two circles are very far from each other, the cross energy can be made arbitrarily small.  
If the linking number  <math>
\mathrm {link}(\gamma_1,\gamma_2)</math> is non-zero, the link is called non-split and for the non-split link, <math>E(\gamma_1,\gamma_2)\geq 4\pi </math>. 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 <math>\mathbb R^n </math>. The Möbius energy has the remarkable property of being invariant under conformal transformations of <math>\mathbb R^3</math>. This property is explained as follows.  Let  <math>F:\mathbb R^3 \rightarrow {S^3}</math> denote a conformal map. Then <math>
E(\gamma_1,\gamma_2) = E(F\circ \gamma_1,F\circ \gamma_2).
</math> This condition  is called the conformal invariance property of the Möbius cross energy.

'''Main Theorem.'''  Let <math>\gamma_i: S^1 \rightarrow \mathbb R^3</math>, <math>i=1,2,</math>  be a non-split link  of 2 components  link.  Then <math>E(\gamma_1,\gamma_2) \geq 2\pi^2</math>. 
Moreover, if <math>E(\gamma_1,\gamma_2) = 2\pi^2</math> then there exists a conformal map <math>F:\mathbb R^3 \rightarrow {S^3}</math> such that <math>F\circ \gamma_1(t)=(\cos t,\sin t,0,0)</math> and <math>F\circ \gamma_2(t)=(0,0,\cos t,\sin t)</math> (the standard Hopf link up to orientation and reparameterization).

Given two non-intersecting differentiable curves <math>\gamma_1, \gamma_2 \colon S^1 \rightarrow \mathbb{R}^3</math>, define the '''[[Carl Friedrich Gauss|Gauss]] map''' <math>\Gamma</math> from the [[torus]] to the [[unit sphere|sphere]] by
:<math>\Gamma(s,t) = \frac{\gamma_1(s) - \gamma_2(t)}{|\gamma_1(s) - \gamma_2(t)|}.</math>
The Gauss map of a link <math>(\gamma_1,\gamma_2)</math> in <math>\mathbf R^4</math>, denoted by <math>g=G(\gamma_1,\gamma_2)</math>, is the Lipschitz map <math>g:S^1 \times S^1 \to S^3</math> defined by
<math>
g(s,t) = \frac{\gamma_1(s)-\gamma_2(t)}{|\gamma_1(s)-\gamma_2(t)|}.
</math>
We denote an open ball in <math>\mathbf R^4</math>, centered at <math>\mathbf x</math> with radius <math>r</math>, by <math>B^4_r(\mathbf x)</math>. The boundary of this ball is denoted by <math>S^3_r(\mathbf x)</math>. An intrinsic open ball  of <math>S^3</math>,  centered at <math>\mathbf p\in S^3</math> with radius <math>r</math>, is denoted by <math>B_r(\mathbf p)</math>.
We have

:<math>\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|}.</math>	

Thus,

:<math>\begin{align}\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{align}
</math>

It follows that for almost every <math>(s,t)\in S^1\times S^1</math>,
<math>
 |{\rm Jac\, }g|(s,t)\leq  \frac{|\dot \gamma_1(s)||\dot\gamma_2(t)|}{|\gamma_1(s)-\gamma_2(t)|^2}.
</math> 
If equality holds at <math>(s,t)</math>, then 
<math>
\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.
</math>

<math>{\mathbf M}(C) \leq \int_{S^1\times S^1} |{\rm Jac\, }g|\,ds\,dt\leq E(\gamma_1,\gamma_2).</math>

If the link <math>(\gamma_1,\gamma_2)</math> is contained in an oriented affine hyperplane with unit normal vector <math>\mathbf p\in S^3</math> compatible with the orientation, then  <math>C={\rm link}(\gamma_1,\gamma_2) \cdot \partial B_{\pi/2}(-\mathbf p).</math>

==References==
* {{Cite book |last=Adams |first=Colin |author-link=Colin Adams (mathematician) |date=2004 |title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots |publisher=American Mathematical Society |isbn=9780821836781
}}
* {{Cite journal |last=Hass |first=Joel |author-link=Joel Hass |title=Algorithms for recognizing knots and 3-manifolds |journal=Chaos, Solitons and Fractals |volume=9 |issue=4–5 |date=April–May 1998 |pages=569–581 |doi=10.1016/S0960-0779(97)00109-4 |arxiv=math/9712269 |bibcode=1998CSF.....9..569H |s2cid=7381505
}}
* {{Cite book |last=Sossinsky |first=Alexei |date=2002 |title=Knots, mathematics with a twist |publisher=Harvard University Press |isbn=9780674009448}}

'''Footnotes'''
{{reflist|refs=
<ref name=OHara1991>{{Cite journal |last=O'Hara |first=Jun |title=Energy of a knot |journal=Topology |volume=30 |issue=2 |date=1991 |pages=241–247 |doi=10.1016/0040-9383(91)90010-2 |doi-access=free |mr=1098918}}</ref>

<ref name=FreedmanHeWang1994>{{cite journal |last1=Freedman |first1=Michael H. |author1-link=Michael Freedman |last2=He |first2=Zheng-Xu |last3=Wang |first3=Zhenghan |title=Möbius energy of knots and unknots |journal=[[Annals of Mathematics]] |series=Second Series |volume=139 |issue=1 |date=January 1994 |pages=1–50 |doi=10.2307/2946626 |mr=1259363 |jstor=2946626}}</ref>

<ref name=Hoste2005>{{cite book |last=Hoste |first=Jim |date=December 2005 |chapter=The enumeration and classification of knots and links |editor1=William W. Menasco |editor2=Morwen B. Thistlethwaite |title=Handbook of Knot Theory |location=Amsterdam |publisher=Elsevier |pages=209–232 |doi=10.1016/B978-044451452-3/50006-X |isbn=9780444514523 |url=http://pzacad.pitzer.edu/~jhoste/HosteWebPages/downloads/Enumeration.pdf}}</ref>

<ref name=AgolMarNeves2012>{{cite arXiv |eprint=1205.0825 |last1=Agol |first1=Ian |author-link1=Ian Agol |last2=Marques |first2=Fernando C. |last3=Neves |first3=André |date=2012 |title= Min-max theory and the energy of links |class=math.GT}}</ref>
}}

{{DEFAULTSORT:Mobius Energy}}
[[Category:Knot theory]]