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  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 selfintersection and ensures the result under gradient descent is of the same knot type. Recall that the Möbius transformations of the 3sphere are the tendimensional group of anglepreserving diffeomorphisms generated by inversion in 2spheres.For example, the inversion in the sphere is defined by , where belongs to .

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  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 selfintersection 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 3sphere are the tendimensional group of anglepreserving diffeomorphisms generated by inversion in 2spheres.For example, the inversion in the sphere is defined by Consider a rectifiable simple curve in the Euclidean 3space , 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 aregularization. It is easy to see that isindependent 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 arclength parameterization. Denote by the length of the curve . Then Let denote a unit circle. We have and consequently, since .

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