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
C1,1
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
S3=R3\cupinfty
\{v\inR3\colon|v-a|=\rho\}
x\toa+{\rho2\over|x-a|2} ⋅ (x-a).
Consider a rectifiable simple curve
\gamma(u)
R3
u
R1
S1
E(\gamma)=\iint\left\{ | 1 | - |
|\gamma(u)-\gamma(v)|2 |
1 | \right\}| | |
D(\gamma(u),\gamma(v))2 |
\gamma | (u)|| |
\gamma |
(v)|dudv,
where
D(\gamma(u),\gamma(v))
\gamma(u)
\gamma(v)
E(\gamma)
\gamma
R3
4
\ell
\gamma
\ell/2 | |
E(\gamma)=\int | |
-\ell/2 |
x+\ell/2 | |
{}dx\int | |
x-\ell/2 |
\left[{1\over|\gamma(x)-\gamma(y)|2}-{1\over|x-y|2}\right]dy.
Let
\gamma0(t)=(\cost,\sint,0)
|\gamma0(x)-\gamma
2={\left(2\sin\tfrac12(x-y)\right) | |
0(y)| |
2}
and consequently,
\begin{align}E(\gamma0)&=\int
\pi | |
-\pi |
x+\pi | |
{}dx\int | |
x-\pi |
\left[{1\over\left(2\sin\tfrac12(x-y)\right)2}-{1\over|x-y|2}\right]dy\\
\pi | |
&=4\pi\int | |
0 |
\left[{1\over\left(2\sin(y/2)\right)2}-{1\over|y|2}\right]dy\\
\pi/2 | |
&=2\pi\int | |
0 |
\left[{1\over\sin2y}-{1\over|y|2}\right]dy\\ &=2\pi\left[{1\overu}-\cot
\pi/2 | |
u\right] | |
u=0 |
=4 \end{align}
since
1u-\cot | |||
|
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.[1] Mathematically, we can say a knot
K
K\colon[0,1]\toR3
K(0)=K(1)
K1,K2
h\colon\R3\to\R3
h(K1)=K2
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. Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is. The special case of recognizing the unknot, called the unknotting problem, is of particular interest.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
k
k+1
c
k
k-1
k+1
k
c
k
k
k+1
k-1
k+1
k
\{k\pm1,k\}
\DeltaK(t)
n
-1 | |
\Delta | |
K(t |
)tn-1=\DeltaK(t)
K
n>0
\DeltaK(t)
t2-t+1
Let
\omega(\boldsymbol{x})= | 1{4\pi}\varepsilon |
ijk |
{xidxj\wedgedxk\over|\boldsymbol{x}|3}
S2
link(\gamma1,\gamma2)=\int\boldsymbol{x\in\gamma1,\boldsymbol{y}\in\gamma2}\omega(\boldsymbol{x}-\boldsymbol{y})
\int | \omega(\boldsymbol{x})= | |
S2 |
1{4\pi}\int | ||||
|
\varepsilonijkxidxjdxk=1, \omega(λ\boldsymbol{x})=\omega(\boldsymbol{x}){\rm{sign}}λ, {\rm{for}} λ\in\R*.
For the knot
\gamma:[0,1] → R3
\gamma(0)=\gamma(1)
\int | |
t1<t2<t3<t4<1 |
\omega(\gamma(t1)-\gamma(t3))\wedge\omega(\gamma(t2)-\gamma(t4))
+\int | |
t1<t2<t3,\boldsymbol{x |
\in\R3\setminus\gamma([0,1])}\omega(\gamma(t1)-\boldsymbol{x})\wedge\omega(\gamma(t2)-\boldsymbol{x})\wedge\omega(\gamma(t3)-\boldsymbol{x})
does not change, if we change the knot
\gamma
Let
\gamma
\R3
T
S3=\R3\cupinfty
T(\gamma)
\R3
E(T(\gamma))=E(\gamma)
T(\gamma)
infty
E(T(\gamma))=E(\gamma)-4
Theorem A. Among all rectifiable loops
\gamma\colonS1\to\R3
E(roundcircle)=4
\gamma
Proof of Theorem A. Let
T
\gamma
E(T(\gamma))\ge0
T(\gamma)
Proof of Möbius Invariance Property. It is sufficient to consider how
I
u
\gamma
u\in\R/\ell\Z
and
Clearly,
E(\gamma)=\lim\varepsilon\to0E\varepsilon(\gamma)
E(I\circ\gamma)=\lim\varepsilon\to0E\varepsilon(I\circ\gamma)
\|I'(\gamma(u))\| ⋅ \|I'(\gamma(v))\| | = | |
|I(\gamma(u))-I(\gamma(v))|2 |
1 | |
|\gamma(u)-\gamma(v)|2 |
.
Since
u
\gamma
Let
s
I\circ\gamma
ds(u)/du=\|I'(\gamma(u))\|
\|I'(\gamma(u))\|=f(u)
I'
\gamma(u)
I'
f(u)
f'(u)\inLinfty
where
L=\rm{Length}(I(\gamma))
\begin{align} \varepsilon+&=\varepsilon+(u)=D((I\circ\gamma)(u),(I\circ\gamma) (u+\varepsilon))=s(u+\varepsilon)-s(u)\\ &=\int
u+\varepsilon | |
u |
1(1-t)f'(u+\varepsilon | |
f(w)dw =f(u)\varepsilon+\varepsilon | |
0 |
t)dt\end{align}
and
\varepsilon-=\varepsilon
1(1-t)f'(u-\varepsilon | |
0 |
t)dt.
Since
|f'(w)|
\begin{align} | 1 | =& |
\varepsilon+ |
1 | |
f(u)\varepsilon |
\left[{1+ {\varepsilon\over
1(1-t)f'(u+\varepsilon | |
f(u)} \int | |
0 |
t)dt}\right]-1\\ =&
1 | \left[1- | |
f(u)\varepsilon |
\varepsilon | |
f(u) |
1(1-t)f'(u+\varepsilon | |
\int | |
0 |
t)dt+
| |||||
O(\varepsilon | - |
1 | |
f(u)2 |
1(1-t)f'(u+ \varepsilon | |
\int | |
0 |
t)dt+ O(\varepsilon). \end{align}
Similarly,
1 | = | |
\varepsilon- |
1 | + | |
f(u)\varepsilon |
1 | |
f(u)2 |
1(1-t)f'(u-\varepsilon t)dt+ | |
\int | |
0 |
O(\varepsilon).
Then by
Comparing and, we get
E\varepsilon(\gamma)-E\varepsilon(I\circ\gamma)=O(\varepsilon);
E(\gamma)=E(I\circ\gamma)
For the second assertion, let
I
\gamma
L=infty
The Freedman–He–Wang conjecture (1994) stated that the Möbius energy of nontrivial links in
R3
\gammai:S1 → R3
i=1,2,
1) | |
\gamma | |
1(S |
\cap
1) | |
\gamma | |
2(S |
=\emptyset
(\gamma1,\gamma2)
E(\gamma1,\gamma2)=
\int | |
S1 x S1 |
| ||||||||||||
|
dsdt.
The linking number of
(\gamma1,\gamma2)
\begin{align} link(\gamma1,\gamma2)&=
1 | |
4\pi |
\oint | |
\gamma1 |
\oint | |
\gamma2 |
r1-r2 | ||||||||||||
|
⋅ (dr1 x dr2)\\ &=
1 | |
4\pi |
\int | |
S1 x S1 |
| ||||||||||||
|
dsdt.\end{align}
… | ||||||
linking number −2 | linking number −1 | linking number 0 | ||||
… | ||||||
linking number 1 | linking number 2 | linking number 3 |
E(\gamma1,\gamma2)\geq4\pi|{\rmlink}(\gamma1,\gamma2)|
link(\gamma1,\gamma2)
E(\gamma1,\gamma2)\geq4\pi
Rn
R3
F:R3 → {S3}
E(\gamma1,\gamma2)=E(F\circ\gamma1,F\circ\gamma2).
Main Theorem. Let
\gammai:S1 → R3
i=1,2,
E(\gamma1,\gamma2)\geq2\pi2
E(\gamma1,\gamma2)=2\pi2
F:R3 → {S3}
F\circ\gamma1(t)=(\cost,\sint,0,0)
F\circ\gamma2(t)=(0,0,\cost,\sint)
Given two non-intersecting differentiable curves
\gamma1,\gamma2\colonS1 → R3
\Gamma
\Gamma(s,t)=
\gamma1(s)-\gamma2(t) | |
|\gamma1(s)-\gamma2(t)| |
.
(\gamma1,\gamma2)
R4
g=G(\gamma1,\gamma2)
g:S1 x S1\toS3
g(s,t)=
\gamma1(s)-\gamma2(t) | |
|\gamma1(s)-\gamma2(t)| |
.
R4
x
r
4 | |
B | |
r(x) |
3 | |
S | |
r(x) |
S3
p\inS3
r
Br(p)
\partialg | |
\partials |
={
\gamma |
1-\langleg,
\gamma |
1\rangleg\over|\gamma1-\gamma
|
=-{
\gamma |
2-\langleg,
\gamma |
2\rangleg\over|\gamma1-\gamma2|}.
Thus,
\begin{align}\left|
\partialg | |
\partials |
\right|2\left|
\partialg | |
\partialt |
\right|2-\left\langle
\partialg | |
\partials |
,
\partialg | |
\partialt |
\right\rangle2 &\leq\left|
\partialg | |
\partials |
\right|2\left|
\partialg | |
\partialt |
\right|2\\ &=
| |||||||||||||||||||
|
| |||||||||||||||||||
|
\\ &\leq
| ||||||||||||
|
. \end{align}
It follows that for almost every
(s,t)\inS1 x S1
|{\rmJac}g|(s,t)\leq
| ||||||||||||
|
.
(s,t)
\langle
\gamma |
|
2(t)\rangle=\langle
\gamma |
1(s),\gamma1(s)-\gamma2(t)\rangle=\langle
\gamma |
2(t),\gamma1(s)-\gamma2(t)\rangle=0.
{M}(C)\leq
\int | |
S1 x S1 |
|{\rmJac}g|dsdt\leqE(\gamma1,\gamma2).
If the link
(\gamma1,\gamma2)
p\inS3
C={\rmlink}(\gamma1,\gamma2) ⋅ \partialB\pi/2(-p).
Footnotes