How nice are critical knots? Regularity theory for knot energies

Tangled Magnetic Fields in Astro- and Plasma Physics; Quantised Flux in Tightly Knotted and Linked Systems

Blatt, Simon^1 ; Reiter, Philipp^2

Workgroup Applied Analysis, Karlsruhe Institute of Technology, Kaiserstraße 89-93, Karlsruhe

76133, Germany^1

Fakultät für Mathematik, Universität Duisburg-Essen, Forsthausweg 2, Duisburg

47057, Germany^2

关键词: Geometric properties;    Knot energy;    Knot theory;    Regularity theories;    Stationary points;    Sub-critical;    Tangent point;    Two parameter;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/544/1/012020/pdf
DOI  :  10.1088/1742-6596/544/1/012020

来源: IOP

【 摘 要 】

In this note we report on some recent developments in geometric knot theory which aims at finding links between geometric properties of a given knotted curve and its knot type. The central object of this field are so-called knot energies which are defined on closed embedded curves. First we present three important examples of two-parameter knot energy families, namely O'Hara's energies, the (generalized) integral Menger curvature, and the (generalized) tangent- point energies. Subsequently we outline the main steps that lead to C-regularity of stationary points- especially minimizers-in the non-degenerate sub-critical range of parameters. Particular attention is devoted to the appearing parallels between these energies which, surprisingly at first glance, are quite similar from an analyst's perspective.