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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 self-intersection and ensures the result under gradient descent is of the same . 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. Conjecturally, there is no energy minimizer for composite knots. 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 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 arc-length parameterization. Denote by the length of the curve . Then Let denote a unit circle. We have and consequently, since . (en)
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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. (en)
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- Ochiai unknot.svg (en)
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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 self-intersection and ensures the result under gradient descent is of the same . 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 and consequently, (en)
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