Relaxation of magnetic knots to braids and groundstate energy minima

In this talk we review some recent progress and present new results on the relaxation of magnetic knots to braids and to their groundstate energy minima. By referring to tubular knot embeddings in ideal magnetohydrodynamics, we illustrate how magnetic knots are generically driven by the Lorentz force to inflexion-free spiral knots and braids [1]. Then, by using classical results by Arnold, Moffatt and Freedman & He, we show that topological crossing number information provides a lower bound on the minimum magnetic energy of knots [2].

By combining recent analytical results on magnetic energy relaxation and data on numerical knot tightening, we establish new relationships between ropelength and groundstate energy minima of knots and links, showing a remarkable similarity between the two spectra. New relationships between ropelength and minimum crossing number are presented and compared with current results [3].

This work provides useful applications for the study of astrophysical flows, and helps to establish a mathematical foundation for the classification of physical knots and links based on a one-to-one correspondence between energy and topology.

[1] Ricca, R.L. (2013) New energy and helicity lower bounds for knotted and braided magnetic fields. Geophys. Astrophys. Fluid Dyn. 107, 385-402.
[2] Ricca, R.L. (2008) Topology bounds energy of knots and links. Proc. R. Soc. A 464, 293-300.
[3] Ricca, R.L. & Maggioni, F. (2014) On the groundstate energy spectrum of magnetic knots and links. J. Phys. A: Math. & Theor., submitted.