The invariance of decomposed Möbius energies under inversions with center on curves (Q2796949)

The authors study the invariance of each part of the Möbius energy under the inversions with respect to the spheres centered on a knot. Recall that the Möbius energy \(\mathcal{E}(f)\) of a closed curve \(f\) in \(\mathbb{R}^n\) can be decomposed in three parts \(\mathcal{E}_1(f),\;\mathcal{E}_2(f)\) and \(4\). They assume that the considered knot has extraregularity. The result holds not only for knots but also for closed curves in \(\mathbb{R}^n\). Assume that \(f\in C^{1,1}\) has bi-Lipschitz continuity.NEWLINENEWLINENEWLINETheorem 1.2: Assume that \(f\in C^{1,1}(\mathbb{R}/\mathcal{L}\mathbb{Z})\) and that there exists a positive constant \(\lambda\) satisfying \(\|f(s_1)-f(s_2)\|_{\mathbb{R}^n}\geq \lambda^{-1}\mathcal{D}(f(s_1),f(s_2))\). Let \(c\) be a point on the curve \(f\), and let \(p\) be an inversion of \(f\) with respect to the sphere whose center is \(c\). Then it follows that \(\mathcal{E}_1(p)=\mathcal{E}_1(f)-2\pi^2\), \(\mathcal{E_2}(p) =\mathcal{E}_2(f)+2\pi^2\).