Symmetric elastic knots

Mathematische Annalen, vol. 385, pp. 811–844

Abstract

Abstract Minimizing the bending energy within knot classes leads to the concept of elastic knots which has been initiated by von der Mosel (Asymptot Anal 18(1–2):49–65, 1998). Motivated by numerical experiments in Bartels and Reiter (Math Comput 90(330):1499–1526, 2021) we prescribe dihedral symmetry and establish existence of dihedrally symmetric elastic knots for knot classes admitting this type of symmetry. Among other results we prove that the dihedral elastic trefoil is the union of two circles that form a (planar) figure-eight. We also discuss some generalizations and limitations regarding other symmetries and knot classes.

Authors 3

1. Alexandra Gilsbach

   Tokyo Institute of Technology

   Affiliation as printed

   Department of Mathematics, School of Science, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan

2. Philipp Reiter corresponding

   Chemnitz University of Technology

   Affiliation as printed

   Chemnitz University of Technology, Faculty of Mathematics, 09107, Chemnitz, Germany

3. Heiko von der Mosel Aachen Institut für Mathematik

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen University, Institut für Mathematik, Templergraben 55, 52062, Aachen, Germany