Energy of a knot: variational principles; Mm-energy

Karpenkov, ON ORCID: 0000-0002-3358-6998 (2004) Energy of a knot: variational principles; Mm-energy. Fund. Math. Today. pp. 214-223.

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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
Additional Information:	17 pages, 6 Postscript figures
Uncontrolled Keywords:	math.GT, math.GT, math-ph, math.MP, 57M25 (Primary); 35A15 (Secondary)
Depositing User:	Symplectic Admin
Date Deposited:	05 Dec 2018 16:19
Last Modified:	19 Jan 2023 01:09
Related URLs:	Author
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3029573