On the groundstate energy spectrum of magnetic knots and links

Abstract

By using analytical results for the constrained minimum energy of magnetic knots we determine the influence of internal twist on the minimum magnetic energy levels of knots and links, and by using ropelength data from the RIDGERUNNER tightening algorithm (Ashton et al 2011 Exp. Math. 20 57–90) we obtain the groundstate energy spectra of the first 250 prime knots and 130 prime links. The two spectra are found to follow an almost identical logarithmic law. By assuming that the number of knot types grows exponentially with the topological crossing number, we show that this generic behavior can be justified by a general relationship between ropelength and crossing number, which is in good agreement with former analytical estimates (Buck and Simon 1999 Topol. Appl. 91 245–57, Diao 2003 J. Knot Theory Ramifications 12 1–16). Moreover, by considering the ropelength averaged over a given knot family, we establish a new connection between the averaged ropelength and the topological crossing number of magnetic knots.