Möbius energy

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 C 1 , 1 {\displaystyle C^{1,1}} 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. 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

S 3 = R 3 ∪ ∞ {\displaystyle S^{3}=\mathbf {R} ^{3}\cup \infty } are the ten-dimensional group of angle-preserving diffeomorphisms generated by inversion in 2-spheres. For example, the inversion in the sphere { v ∈ R 3 : | v − a | = ρ } {\displaystyle \{\mathbf {v} \in \mathbf {R} ^{3}\colon |\mathbf {v} -\mathbf {a} |=\rho \}} is defined by

x → a + ρ 2 | x − a | 2 ⋅ ( x − a ) . {\displaystyle \mathbf {x} \to \mathbf {a} +{\rho ^{2} \over |\mathbf {x} -\mathbf {a} |^{2}}\cdot (\mathbf {x} -\mathbf {a} ).}

Consider a rectifiable simple curve γ ( u ) {\displaystyle \gamma (u)} in the Euclidean 3-space R 3 {\displaystyle \mathbf {R} ^{3}} , where u {\displaystyle u} belongs to R 1 {\displaystyle \mathbf {R} ^{1}} or S 1 {\displaystyle S^{1}} . Define its energy by

E ( γ ) = ∬ { 1 | γ ( u ) − γ ( v ) | 2 − 1 D ( γ ( u ) , γ ( v ) ) 2 } | γ ˙ ( u ) | | γ ˙ ( v ) | d u d v , {\displaystyle E(\gamma )=\iint \left\{{\frac {1}{|\gamma (u)-\gamma (v)|^{2}}}-{\frac {1}{D(\gamma (u),\gamma (v))^{2}}}\right\}|{\dot {\gamma }}(u)||{\dot {\gamma }}(v)|\,du\,dv,}

where D ( γ ( u ) , γ ( v ) ) {\displaystyle D(\gamma (u),\gamma (v))} is the shortest arc distance between γ ( u ) {\displaystyle \gamma (u)}

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