On the minimizers of the Möbius cross energy of links

Abstract

We give a geometric interpretation for the Euler-Lagrange equation for the Möbius cross energy of (nontrivially linked) 2-component links in the euclidean 3-space. The minimizer of this energy is conjectured to be a Hopf link of 2 round circles. We prove some elementary properties of the minimizers using the Euler-Lagrange equations. In particular, we give a rigorous proof of the fact that the minimizer is topologically a Hopf link.