Möbius energy

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 Freedman, He, and Wang (1994) who used it to show the existence of a C1,1 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. Kusner and 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[edit]

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.

(copied content from Knot equivalence; see that page's history for attribution.)

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 (Adams 2004) (Sossinsky 2002). 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 as an 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 (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 (Hass 1998). The special case of recognizing the unknot, called the unknotting problem, is of particular interest (Hoste 2005). 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.

The representations of Temperley-Lieb-Jones algebras[edit]

We shall be interested in the von Neumann algebras generated by 1 and projectors which obey :

These algebras in addition admit of a trace, denoted by "tr" which is defined over and determined by the normalization tr (1) = 1 and the following conditions :

Jones has obtained a representation of this algebra and a corresponding representation of the braid group. The braid group Bn can be abstractly defined via the following presentation:

where in the first group of relations 1 ≤ i ≤ n−2 and in the second group of relations, |i − j| ≥ 2. This presentation leads to generalisations of braid groups called Artin groups. The cubic relations, known as the braid relations, play an important role in the theory of Yang–Baxter equation.

Möbius Invariance Property. 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 iff 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

and

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

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 .

where and

and

Since is uniformly bounded, we have

Similarly,

Then by (4)

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[edit]

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.[1] 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.svg Linking Number -1.svg Linking Number 0.svg
linking number −2 linking number −1 linking number 0
Linking Number 1.svg Linking Number 2.svg Linking Number 3.svg
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[edit]

  1. ^ Agol, Ian; Marques, Fernando C.; Neves, André (2012). "Min-max theory and the energy of links". arXiv:1205.0825Freely accessible [math.GT].