On symmetric critical points of knot energies

• Über Symmetrische Kritische Punkte von Knotenenergien

Gilsbach, Alexandra Philippa; von der Mosel, Heiko (Thesis advisor); Strzelecki, Pawel (Thesis advisor)

Aachen (2018)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2018

Abstract

In this thesis, we examine the energy landscape of knot energies, trying to gain information about whether in a knot class there is more than one critical point. Motivated by the work of Cantarella et al. we use the principle of symmetric criticality of Palais to show the existence of several symmetric critical points in torus knot classes for three families of continuously differentiable knot energies: the Integral Menger curvature M_p, the Tangent Point energies T(r,q) and the O’Hara energies E_alpha. A tool we need is that a fixed torus knot may not have two or more relative prime periods, i.e. rotational symmetries where the knot is disjoint from the axis of rotation. We give a geometrical proof for this. We present experiments with a numerical gradient flow for M_p established by Hermes to gain numerical evidence on whether the symmetric critical points are stable critical points or saddle points. The numerical results suggest that minimisers of M_p convergeto minimisers of the Ropelength functional. We give an analytical proof for this, showing that M_p Gamma-converges to Ropelength for p tending to infinity. With this result we are able to prove the convergence of minimisers and symmetric critical points of M_p to those of Ropelength.