Symmetric elastic knots

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.