Induction effects of torus knots and unknots

Abstract

The induction effects due to a steady source field in the shape of a torus knot or unknot filament are analysed in detail. Similar studies for rectilinear, circular or helical geometries have been done in the past, but very little is known for more complex geometries and topologies. Torus knots provide a rare example of closed, space curves of non-trivial topology, that admit a mathematically simple description; for this reason they represent an interesting case study to consider. Moreover, since torus knots are also a good mathematical model for studying braided field line structures, the present work provides useful information for a wide range of possible applications, from physical sciences (solar physics and astrophysics, vortex dynamics, fusion physics) to technology (telecommunication, new materials design, data analysis). The work is organized in 4 chapters.In chapter 1 we present a comprehensive study of geometric and topological properties of torus knots and unknots. By using a standard parametrization, we demonstrate the existence, and determine the location, of inflection points for a given critical configuration, and prescribe the condition for removing the singularity associated with torsion at the inflection point. We show that, to first approximation, total length grows linearly with the number of coils, and it is proportional to the minimum crossing number of the knot type. By taking the winding number, given by the ratio between meridian and longitudinal wraps, as measure of topological complexity of the knot, we analyse its influence on several global quantities, such as total length, curvature, torsion and writhe.In chapter 2 we analyse the influence of the winding number and other geometric properties on induction, energy and helicity. This is done by assuming the physical filament of infinitesimally small cross-section and by using the Biot-Savart law adapted for the particular parametrization chosen. Field line patterns of the induced field are obtained for a large family of knots/unknots on several crosssectional planes. The intensity of the induced field is shown to depend linearly on the number of toroidal coils. We provide bounds on energy, and an estimate of helicity in terms of writhe.In chapter 3 we compare local and global induction contributions in relation to the winding number, by providing asymptotic expansions of the integrand function. We show that in general local leading order terms are not sufficient to provide accurate global information; nevertheless, for some values of the winding number local and global behaviours are found to be in good agreement.In chapter 4 we investigate the influence of the winding number on the binormal component of the self-induction a point asymptotically near to the source field. Since in the limit the Biot-Savart integral becomes singular, we apply the analytical prescription of Moore and Saffman (1972) to regularize it. While to leading order the self-induction is proportional to local curvature, we derive an integral formula for nex...

Influence of winding number on vortex knots dynamics

“…The translation speed U can be computed explicitly for particular knot configurations. For the family ( q = 3, 5, …, 15), let us consider the algebraic mean of the induced velocities at various points in the meridian plane 33 . The normalized speed is thus defined by…”

Knots cascade detected by a monotonically decreasing sequence of values

“…The sequence is clearly independent from the specific physical context and it appears to be consistent with the typical behavior of global geometric properties of standard torus knots. Computations of various quantities, such as first-order elastic energy (bending and torsional) of thin rods 22 and kinetic energy of vortex filaments 23 (see inset of Fig. 4 ) demonstrate a remarkable similarity with the monotonically decreasing sequence of HOMFLYPT values.…”

“…of type shown in Fig. 2 , with all over-crossings); for this knot family, denoted by { T (2, n )} + reference values for Wr and Tw must be taken between 0 and 1 22 . For the family { T (2, n )} − of mirror knots with all under-crossings, i.e.…”

“…For negative T (2, n ) reference values for Wr and Tw must be taken between −1 and 0 22 . Note that by using (13) and by replacing τ → − τ and ω → − ω , equation (12) remains unchanged.…”

“…HOMFLYPT computation given by (12) for positive T (2, n ), with λ ω = λ τ = 1/2, | Wr | = | Tw | = 1/2 and n = 4, 5, …, 10: for decreasing n we obtain a monotonically decreasing sequence of numerical values (circles denote exact values, the dashed curve represents the best fit given by a quartic polynomial). Inset: typical behavior of global geometric properties of positive torus knots: writhe Wr , total squared curvature , total curvature K and total length (over-bars denote non-dimensional quantities) 22 .…”