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.Keywords: ideal magnetohydrodynamics, magnetic knots and links , magnetic energy spectrum, tight knots and links, ropelength PACS numbers: 02.10. Kn, 47.10.A, 52.30.Cv (Some figures may appear in colour only in the online journal) Magnetic knots and links in an ideal fluidThe search for possible relationships between energy and topology has a long history, which has its roots in Lord Kelvin's vortex atom theory and Tait's knot tabulation (see, for instance,

Minimal unlinking pathways as geodesics in knot polynomial space

“…7 is extrapolated from purely topological data, without any reference to physical or biological processes. A physical justification of this logarithmic growth, however, can be given by comparing these results with data obtained from the analysis of minimum energy states of magnetic knots and links 21 and, to some extent, of elastic systems 22 . By examining the numerical values of the groundstate spectra determined there we can see that energy minima grow with topological complexity as a ln # K þ b, with a and b constants (and independent of knot types), and # K denoting knot tabulation according to increasing ropelength λ.…”

“…It is also worth noticing that the best-fit curve of the optimal geodesic given by the most probable sequence Π 1 is arguably the closest match to the best-fit behavior obtained in ref. 21 . In summary, a new way to analyze and interpret optimal pathways attained through topological simplification of physical knots and links by step-wise sequence of unlinking events, such as those studied by Stolz et al 2 , is introduced and tested.…”

“…We find that the data obtained by our approach match very well the data obtained by the other method, suggesting that the alternative method based on use of geodesics in knot polynomial space has great potential for probability computation of decaying processes of physical systems, even far more complex than the one considered here. By using a new measure of topological complexity based on distances in the knot space we also show that the decaying patterns observed are captured by a logarithmic best-fit curve that is functionally related, and bounded from below, by the logarithmic distribution of minimum energy states of tight physical knots 21,22 .…”

“…Since we refer to physical knots, we make use of the adapted polynomials introduced by Liu and Ricca 18,19 to quantify the topology of fluid knots.This is done by taking standard Legendre polynomials to construct a basis and an appropriate metric to compute distance using knot polynomials. This allows to measure and compare geodesic distances between knot types, define and compute relative probabilities, and relate minimal pathways to lower bounds given by minimum energy states [20][21][22] .…”

Ropelength, crossing number and finite type invariants of links

“…[23], [24]. Bounds for the ropelength of knots, and in particular the lower bounds, have been studied by many researchers, we only list a small fraction of these works here [5,6,9,14,13,12,16,31,39,40,32,41]. Many of the results are applicable directly to links, but the case of links is treated in more detail by Cantarella, Kusner and Sullivan [9] and in the earlier work of Diao, Ernst, and Janse Supported by NSF DMS 1043009 and DARPA YFA N66001-11-1-4132 during the years 2011-2015.…”