Symmetric elastic knots

TL;DR

This paper proves the existence of dihedrally symmetric elastic knots within certain knot classes, showing, for example, that the elastic trefoil forms a planar figure-eight, and discusses symmetry-related limitations.

Contribution

It establishes the existence of dihedrally symmetric elastic knots and characterizes the elastic trefoil as a union of two circles forming a figure-eight.

Findings

Dihedral elastic trefoil is a planar figure-eight.

Existence of dihedrally symmetric elastic knots proven.

Discussion of symmetry limitations in elastic knots.

Abstract

Minimizing the bending energy within knot classes leads to the concept of elastic knots which has been initiated in [von der Mosel, Asymptot. Anal. 1998]. Motivated by numerical experiments in arxiv:1804.02206 (doi:10.1090/mcom/3633) 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.