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

Consider a rectifiable simple curve in the Euclidean 3-space , where belongs to or . Define its energy by

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

Let denote a unit circle. We have

and consequently,

since .

Knot invariant

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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   is an injective and continuous function   with  . 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   are equivalent if there is an orientation-preserving homeomorphism   with  , 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   to the region number  . Clearly, at any crossing point  , there are two opposite corners of the same number   and two opposite corners of the numbers   and  , respectively. The number   is referred as the index of  . 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   two dotted corners are of numbers   and  , respectively, two undotted ones of numbers   and  . The index of any corner of any region of index   is one element of  . 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   with integer coefficient. The Alexander polynomial is symmetric with degree  :   for all knots   of   crossing points. For example, the invariant   of an unknotted curve is 1, of an trefoil knot is  .

Let

  denote the standard surface element of  .

We have

 
 

For the knot  ,  ,

 
 

does not change, if we change the knot   in its equivalence class.

Möbius Invariance Property

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Let   be a closed curve in   and   a Möbius transformation of  . If   is contained in   then  . If   passes through   then  .

Theorem A. Among all rectifiable loops  , round circles have the least energy   and any   of least energy parameterizes a round circle.

Proof of Theorem A. Let   be a Möbius transformation sending a point of   to infinity. The energy   with equality holding if and only if   is a straight line. Apply the Möbius invariance property we complete the proof.

Proof of Möbius Invariance Property. It is sufficient to consider how  , an inversion in a sphere, transforms energy. Let   be the arc length parameter of a rectifiable closed curve  ,  . Let

  (1)

and

  (2)

Clearly,   and  . It is a short calculation (using the law of cosines) that the first terms transform correctly, i.e.,

 

Since   is arclength for  , the regularization term of (1) is the elementary integral

  (3)

Let   be an arclength parameter for  . Then   where   denotes the linear expansion factor of  . Since   is a Lipschitz function and   is smooth,   is Lipschitz, hence, it has weak derivative  .

  (4)

where   and

 

and

 

Since   is uniformly bounded, we have

 

Similarly,  

Then by (4)

  (5)

Comparing (3) and (5), we get   hence,  .

For the second assertion, let   send a point of   to infinity. In this case   and, thus, the constant term 4 in (5) disappears.

Freedman–He–Wang conjecture

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The Freedman–He–Wang conjecture (1994) stated that the Möbius energy of nontrivial links in   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  ,   be a link of 2 components, i.e., a pair of rectifiable closed curves in Euclidean three-space with  . The Möbius cross energy of the link   is defined to be

 

The linking number of   is defined by letting

 
       
linking number −2 linking number −1 linking number 0
       
linking number 1 linking number 2 linking number 3

It is not difficult to check that  . If two circles are very far from each other, the cross energy can be made arbitrarily small. If the linking number   is non-zero, the link is called non-split and for the non-split link,  . 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  . The Möbius energy has the remarkable property of being invariant under conformal transformations of  . This property is explained as follows. Let   denote a conformal map. Then   This condition is called the conformal invariance property of the Möbius cross energy.

Main Theorem. Let  ,   be a non-split link of 2 components link. Then  . Moreover, if   then there exists a conformal map   such that   and   (the standard Hopf link up to orientation and reparameterization).

Given two non-intersecting differentiable curves  , define the Gauss map   from the torus to the sphere by

 

The Gauss map of a link   in  , denoted by  , is the Lipschitz map   defined by   We denote an open ball in  , centered at   with radius  , by  . The boundary of this ball is denoted by  . An intrinsic open ball of  , centered at   with radius  , is denoted by  . We have

 

Thus,

 

It follows that for almost every  ,   If equality holds at  , then  

 

If the link   is contained in an oriented affine hyperplane with unit normal vector   compatible with the orientation, then  

References

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  • 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

  1. ^ O'Hara, Jun (1991). "Energy of a knot". Topology. 30 (2): 241–247. doi:10.1016/0040-9383(91)90010-2. MR 1098918.
  2. ^ Freedman, Michael H.; He, Zheng-Xu; Wang, Zhenghan (January 1994). "Möbius energy of knots and unknots". Annals of Mathematics. Second Series. 139 (1): 1–50. doi:10.2307/2946626. JSTOR 2946626. MR 1259363.
  3. ^ Adams 2004; Sossinsky 2002.
  4. ^ a b Hass 1998.
  5. ^ Hoste, Jim (December 2005). "The enumeration and classification of knots and links". In William W. Menasco; Morwen B. Thistlethwaite (eds.). Handbook of Knot Theory (PDF). Amsterdam: Elsevier. pp. 209–232. doi:10.1016/B978-044451452-3/50006-X. ISBN 9780444514523.
  6. ^ Agol, Ian; Marques, Fernando C.; Neves, André (2012). "Min-max theory and the energy of links". arXiv:1205.0825 [math.GT].