Influence of winding number on vortex knots dynamics

In this paper we determine the effects of winding number on the dynamics of vortex torus knots and unknots in the context of classical, ideal fluid mechanics. We prove that the winding number — a topological invariant of torus knots — has a primary effect on vortex motion. This is done by applying the Moore-Saffman desingularization technique to the full Biot-Savart induction law, determining the influence of winding number on the 3 components of the induced velocity. Results have been obtained for 56 knots and unknots up to 51 crossings. In agreement with previous numerical results we prove that in general the propagation speed increases with the number of toroidal coils, but we notice that the number of poloidal coils may greatly modify the motion. Indeed we prove that for increasing aspect ratio and number of poloidal coils vortex motion can be even reversed, in agreement with previous numerical observations. These results demonstrate the importance of three-dimensional features in vortex dynamics and find useful applications to understand helicity and energy transfers across scales in vortical flows.


Reduction of the Biot-Savart Induction Law for Torus Knots and Unknots
Torus knots and unknots. Torus knots and unknots are rotationally symmetric closed curves standardly embedded on a mathematical torus Π in 3  (see 23 ). Each torus knot is defined by a pair of co-prime integers p > 1 and q > 1, denoting the number of full turns around Π done by the curve along the longitudinal (or toroidal) direction and meridian (or poloidal) direction, respectively (see Fig. 1a). The unknot is given by either p = 1 or q = 1 (see Fig. 1b). When p = q = 1 the unknot is a twisted circle on Π, and when p and q are rational, but not relatively prime, we have links (with number of components given by the greatest common divisor between p and q). The ratio w = q/p (w > 0) defines the winding number and is a measure of the knot topology. When w = q/p is irrational the curve forms a dense set covering Π almost everywhere. Two limits are of interest: (i) a poloidal hollow ring covered by infinitely many poloidal coils, when p is fixed and finite, and q → ∞; (ii) a toroidal hollow ring, when q is fixed and finite, and p → ∞.

Biot-Savart law for vortex knots: asymptotic formula and leading order terms. A vortex torus
knot is a vortex filament of negligible cross-section identified with the knot type p q ,  . We take the vortex to be embedded in an inviscid, irrotational, incompressible and unbouded fluid at rest at infinity, with vorticity ω = ∇ × u (u velocity) localized on a thin tube centred on  p q , . The Lagrangian L of the system is defined by L = E − U, where E is the total energy (kinetic energy K plus potential energy U), so that for an isolated vortex in an unbouded fluid it simply coincides with the kinetic energy of the vortex, given by where V is fluid volume. In absence of other contributions, the velocity u is solely that induced by vorticity through the inverse of the curl operator, given by the classical Biot-Savart law 24 as |x| → ∞, the integral (2) converges. Here we want to determine the self-induced motion of the vortex filament in steady conditions. For this we take vorticity ω = ϖ 0t , with ϖ 0 = constant and t unit tangent to  p q , , with Frenet frame given by unit tangent, normal and binormal {t n b , ,ˆˆ} on p q ,  . For the assumptions made above the veloc- ) at any irrotational point  ∈ x 3 exterior to the vortex can thus be written in terms of a line integral, given by where Γ is vortex circulation. The Hamiltonian (per unit density and appropriately regularised) associated with Eq. (3) is given by 25,26   ; these two knots are topologically equivalent to each other, that is one can be deformed to the other by a series of continuous deformations. (b) Poloidal coil  1,5 (left) and toroidal coil 5,1  ; these are unknots topologically equivalent to the standard circle. where integration on α is extended to the number p of toroidal coils. The propagation velocity of the vortex is given by considering the self-induction at a point x o asymptotically close to the nearest source point ⁎ x , i.e. when x = x o → x ⁎ . The asymptotic behavior of (3) was originally derived by Da Rios in 1911, re-formulated by Levi-Civita in 1932 (for a historical reconstruction see 27 ) and subsequently re-discovered by Batchelor 24 . In general, for any point asymptotically close to the vortex the self-induced velocity is given by here ρ is the local radius of curvature, σ the radius of the vortex circular cross-section (with  σ ρ), q the azimuthal unit vector, F some function of the local vorticity distribution, and G a finite term contribution that depends on far-field effects. Since the azimuthal term does not contribute to the displacement of the vortex, the drift velocity is given by v( It is well-known that when x = x o → x * the binormal component gives rise to a logarithmic singularity and, to leading order, is responsible for the drift velocity of the vortex. It is therefore important to focus first on this contribution alone. From (6) we have: where C = [Γ/(4πρ)]F + G · b is now function of the sole geometry and topology of  p q , . Thus, we havê

Analytical De-Singularization of the Biot-Savart Integral
In order to compute C = C(λ, w) from Eq. (8) we must first deal with the analytic logarithmic singularity that arises when x o → x * . Since this singularity has no physical justification, but it is purely an artifact of the analytical expression (3), a de-singularization technique must be applied. For this we follow the prescription of Moore-Saffman 21 . This is based on the application of an asymptotic technique that matches the local shape and vorticity distribution of the given vortex with those of the osculating vortex ring of same local curvature c = ρ −1 and vorticity so to reproduce locally the correct dynamics. It relies on the observation that the streamline pattern induced by the local vortex geometry and core structure is, to leading order, the same as that obtained by replacing a strand of the original vortex with that of a vortex ring of same curvature and core structure. Indeed one can prove 28,29 that by direct application of this method the error is at most of the order O(δ 2 ), where δ = (σ/R)ln(8R/σ) and R / 1 σ  . The singularity is thus removed by virtually subtracting and adding the contribution to the induced velocity at x o , replacing the original vortex with that of the osculating vortex ring o  at x o (see Fig. 2). By exploiting the rotational symmetry of torus knots we consider the equatorial point  Let us calculate the parametric equations of o  . The Frenet frame at x o is given bŷ^Â the vector position of points on the osculating circle  o is given by . By substituting the expressions above into (9), we have

Winding Number Effects on the Self-Induced Velocity
We can now determine the influence of the winding number on the self-induced motion of the vortex. Using (10) and by renaming ϑ to α, the binormal component is given by π π π BS BS BS Binormal component of the self-induced velocity. From (20) and (10) and by comparing the above with (8), we obtainρ Contributions to the binormal component C = C(λ, w) of 42 torus knots and 14 unknots have been examined for various aspect ratios and increasing values of w. The plots are shown in Fig. 3 for (a) λ = 0.25, (b) λ = 0.5, (c) λ = 0.75 and R = 1. Plots on the left show the contributions from toroidal knots and unknots (p > q, w < 1); plots on the right show the contributions from poloidal knots and unknots (q > p, w > 1). As we see C tends to increase with λ, regardless of the knot type; for a given aspect ratio, C generically increases with decreasing number of toroidal coils and increasing number of poloidal coils. From Eq. (23) we notice that this represents only part of the total contribution to the binormal component, since winding number effects are also present in the first term of Eq. (23) (see the influence of w on local curvature in 23 ).

Tangential and normal components of the self-induced velocity.
As mentioned before the tangential and normal contributions to the velocity are not singular. Hence, replacing b into (16) first by t and then by n we have BS k t and k n BS , that after integration over [0, 2πp] give respectively v t (x o ) and v n (x o ). The contribution C t to the tangential component of the velocity at x o is given by considering the tangential component of Eq. (6); combining this latter with Eq. (15), where we replace BS k with BS k t , we have

BS
Plots of C t = C t (λ, w) against different values of aspect ratio and winding number are shown in Fig. 4. As we see |C t | generically increases with increasing p and q, leading to an increase of orbital motion of the knot around the torus central and circular axes. By direct inspection we can also see that k n BS is anti-symmetric with respect to the interval of integration, i.e.
Global contributions to the velocity. The vortex translates in the fluid in the direction of the torus central axis with a speed U, and it is also subject to a combined action of a rigid rotation Θ around the z-axis and an orbital rigid rotation Ω in the meridian direction around the torus circular axis R = 1. Θ and Ω do not contribute to displace the vortex center of mass, and the direction of these contributions is reversed if we reverse the orientation of vorticity, i.e. if we change α → −α in (1). Since torus knots are chiral knots 23 , by changing z * → −z * in (1) left-handed knots are transformed into right-handed knots. Chirality implies a reversal of Ω, but it has no effect on the directions of Θ and U.   www.nature.com/scientificreports www.nature.com/scientificreports/ we see from the plots C z may change sign for particular values of winding number and aspect ratio. It's easy to determine whether this decrease may lead to a local standstill, or even a reverse motion; from (28), we havê^^⟺v

Translation speed. Let us consider the translation speed
This provides an analytical condition for the reversal of the translation velocity of vortex knots and unknots; in particular, as we see from the plots of the poloidal coils of Fig. 5 (with either p = 1 or q = 1), we have proof of the anomalous translation of perturbed vortex rings observed numerically in superfluids 31 ; the results of Fig. 5 for p = 1 and increasing q show that indeed the reversed speed increases with aspect ratio (that corresponds to wave amplitude in Fig. 1(d) of 31 ), and for a given aspect ratio it increases with the number q of poloidal coils (that corresponds to the number of waves in Fig. 2 of 31 ). Our results provide also mathematical grounds to the observed reversed motion of viscous vortex rings in presence of intense swirl 32 . Since swirling flow is generated by a bundle of poloidal vortex lines confined to a ring, high swirl corresponds to high number of poloidal coils (with p = 1), leading to the condition (32) discussed above.
The translation speed U can be computed explicitly for particular knot configurations. For the family q 2,  (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 U is thus defined bȳ¯= Numerical integration at the innermost points presents some difficulty due the appearance of numerical instabilities associated with the concentration of the streamlines in the toroidal region 33 . Plots of U = U(w) are shown in Fig. 6 for extremely thin vortex cores. As was noted before 8,20 the vortex speed increases with the number of toroidal coils p, but here we notice also secondary effects due to q. The order of magnitude of U is only slightly influenced by q when λ ≥ 0.5, but it changes considerably when λ ≤ 0.3. This behavior went unnoticed in 8 . Partial comparison with results obtained in superfluids can be made by taking U o =U/lnδ, with λ = 0.1, σ = 10 −8 and ln δ= 18.61 (typical values for superfluid vortices). Computations of U o = U o (w) for  q 2, (q = 3, 5, 7, 9) are shown in Fig. 7. Direct comparison with data obtained by 20 (Fig. 5,  q 2, for q = 3, 5, 7, 9) is limited by the assumption of superfluid hollow vortex core. The general trend of a decrease in propagation speed as function of  5,7,9,11,13,15); p, 3  and q 3,  (p, q = 4, 5, 7, 8, 10, 11, 13);  p,4 and  q 4, (p, q = 5, 7,9,11,13,15,17). Interpolation is for visualization purposes only.  www.nature.com/scientificreports www.nature.com/scientificreports/ w is confirmed, but in the present case of uniform vorticity core distribution we observe a much more drastic decrease of U o for increasing w. This result is consistent with higher values of rotational energy that at high q induce a much stronger toroidal jet inside Π, with a consequential further reduction of the overall speed of the vortex.

Concluding Remarks
By using the Moore-Saffman de-singularization technique we have studied the self-induced motion of vortex torus knots p q ,  under the full Biot-Savart law. This has been done by analyzing 56 different knots and unknots up to 51 crossings, determining the precise relationships between winding number and velocity contributions. We can state that the influence of winding number has leading order effects on vortex motion, comparable to curvature effects. The effects of w = q/p are related to the relative number of toroidal and poloidal coils. For complex knots the number of toroidal coils p is in general of primary importance. For given p and q knots travel faster than unknots, and if p > q p q ,  knots travel faster than  q p , knots. In particular we prescribe the condition for a reversal of the translation speed, providing theoretical grounds for the numerically observed reversal of vortex rings subject to superimposed large-amplitude perturbations 31 or swirl 32 . We can also establish an increased influence of poloidal coils on the propagation of thin-cored vortex knots that went unnoticed in previous numerical works 8 or that resulted in a much milder effects for superfluid hollow vortex cores 20 .
These new findings help to interpret the evolution of three-dimensional, localized bundle of vortex lines 34,35 in complex networks of vortical flows, where the coiling of field lines has a primary effect on the motion. Indeed, whereas the pitch of an isolated helix was found to have only a secondary effect on vortex motion 22 , the helical winding of vortex lines, due to the combined effects of poloidal and toroidal coils through w (see Eq. (12) above), can actually be of primary importance. Since in reconnection processes twist effects are important for helicity considerations and energy transfers across scales 36,37 , these results are important to understand the role of three-dimensional features in the evolution and coherence of localized vortical flows.