# Energy of a knot: variational principles; Mm-energy

Karpenkov, ON ORCID: 0000-0002-3358-6998
Energy of a knot: variational principles; Mm-energy. Fund. Math. Today. 214 - 223.

This is the latest version of this item.

 Text 0411060v1.pdf - Submitted Version Download (189kB)

## Abstract

Let $E_f$ be the energy of some knot $\tau$ for any $f$ from certain class of functions. The problem is to find knots with extremal values of energy. We discuss the notion of the locally perturbed knot. The knot circle minimizes some energies $E_f$ and maximizes some others. So, is there any energy such that the circle neither maximizes nor minimizes this energy? Recently it was shown (A.Abrams, J.Cantarella, J.H.G.Fu, M.Ghomu, and R.Howard) that the answer is positive. We prove that nevertheless the circle is a locally extremal knot, i.e. the circle satisfies certain variational equations. We also find these equations. Finally we represent Mm-energy for a knot. The definition of this energy differs with one regarded above. Nevertheless besides its own properties Mm-energy has some similar with M\"obius energy properties.

Item Type: Article 17 pages, 6 Postscript figures math.GT, math.GT, math-ph, math.MP, 57M25 (Primary); 35A15 (Secondary) Symplectic Admin 05 Dec 2018 16:19 15 Jul 2019 13:11 Author http://livrepository.liverpool.ac.uk/id/eprint/3029573