L energia minima dei nodi. Francesca Maggioni

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1 L energia minima dei nodi Francesca Maggioni Department of Management, Economics and Quantitative Methods, University of Bergamo (ITALY) Università Cattolica del Sacro Cuore di Brescia, 17 Maggio 2016, Brescia

2 Filamentary physical systems Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

The filament model The filament model The filament F is modelled by a thin inexestensible rod of constant length L and uniform circular cross-section of area A = πa 2 with a/l 1.

3 The filament model The filament model The filament F is modelled by a thin inexestensible rod of constant length L and uniform circular cross-section of area A = πa 2 with a/l 1. The filament axis C is given by a simple, smooth space curve X(s). Each fiber C is given by X (s) = X(s)+ǫˆN(s) with ˆN(s) = ˆncosϑ(s)+ ( ˆbsinϑ(s). X, ˆN ) defines a ribbon of edges X and X. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

The filament model Writhing number Let C be a simple closed, smooth curve in R 3, X(s) : [0,L] R 3 ; Definition (Fuller 1971) The writhing number is defined by Wr(C) 1 ˆt(s) ˆt(s )

4 The filament model Writhing number Let C be a simple closed, smooth curve in R 3, X(s) : [0,L] R 3 ; Definition (Fuller 1971) The writhing number is defined by Wr(C) 1 ˆt(s) ˆt(s ) [X(s) X(s )] ds ds. 4π X(s) X(s ) 3 C C Interpretation of W r in terms of crossing number: Wr(C) =< n +( ν ) n ( ν ) >, Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

5 The filament model Total twist number A measure of the winding of each infinitesimal fiber around C is given by: Definition (Love 1944) The total twist number is defined by Tw = 1 (ˆN ˆN(s)) (s) ˆt(s)ds = T +N, 2π C where T is the normalized total torsion and N the intrinsic twist. Theorem (Călugăreanu, 1959 and White, 1969) Lk = Wr +Tw, ( where Lk is the linking number of the ribbon X, ˆN ). Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

6 Magnetic Knots and Links Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

7 Magnetic Knots and Links Magnetic knots as tubular embeddings in ideal fluid We consider a magnetic field B defined in a tube T = K S: B { B = 0, tb = (u B), L 2 norm}. The Magnetic Energy M is M = 1 2 V (T ) The Bending Energy E b is given by E b = 1 2 K B 2 d 3 x. [c(s)] 2 ds The Magnetic Signature (V(K),Φ = S B d2 x) is constant. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

8 Magnetic Knots and Links Groundstate energy spectra In general we have (Moffatt, 1990): M min(h) = m(h)φ 2 V 1/3 where m(h) is a positive, dimensionless function of the initial twist h; Problem (Moffatt, 2001): Determine m min for knots with topological crossing number c min = 3,4,5... For zero-framed knots (Ricca, 2008) m(0) = (2/π) 1/3 c min; thus M min(0) c min. c min # knots types Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

9 Magnetic Knots and Links The Rolfsen knots table (up to 8 crossings ) R. Scharein Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

10 Magnetic Knots and Links Curvilinear coordinate system (Mericer 1963) A point P on S is given by: x = X(s)+rcosϑˆn(s)+rsinϑˆb(s), ϑ R = ϑ+ s 0 τ (η)dη. (r,ϑ R,s): orthogonal, curvilinear zero-twist coordinate system. dx dx = (dr) 2 +r 2 (dϑ R) 2 +[1 crcos(ϑ R +ϑ )] 2 (ds) 2, metric }{{} k (êr,ê ϑr,ˆt ) : Orthonormal basis in radial, meridian and longitudinal dir. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

11 Magnetic Knots and Links Minimization of magnetic energy for flux tubes Theorem (Maggioni & Ricca, 2009) Let K be an essential magnetic knot in an ideal fluid, with signature {V,Φ} and magnetic field B. We assume that: (i) {V,Φ} is invariant; (ii) The circular cross-section S remains constant along K; (iii) The knot length L is independent of the knot framing h; Minimization of magnetic energy, constrained by (i)-(iii), yields ( ) M L 2 = 2V + πh2 Φ 2. L In term of the ropelength λ = L/R (setting V = πr 2 L = 1 and Φ = 1): M λ(h) = λ4/3 2π 2/3 + π4/3 h 2 λ 2/3. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

12 Magnetic Knots and Links Constrained groundstate energy of prime knots up to 10 crossings by Ridgerunner algorithm (Cantarella, Rawdon, Piatek) Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

13 Knot Energy Spectra: Magnetic vs Bending Energy Knots Energy Spectrum: normalized magnetic energy vs. ropelength V = 1, Φ = 1, h = 0 tight torus: M o = ( 2π 2) 1/3 m = M λ(0) M o = ( λ 2π )4/3 Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

14 Knot Energy Spectra: Magnetic vs Bending Energy Knots Energy Spectrum: normalized bending energy vs. ropelength E b = 1 2 K [c(s)]2 ds tight torus: E o = πr = 2 1/3 π 5/3 ẽ = E b K = [c(s)]2 ds E o 2 4/3 π 5/3 Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

15 Knot Energy Spectra: Magnetic vs Bending Energy Knots Energy Spectrum: ratio between normalized magnetic energy and normalized bending energy Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

16 Links Energy Spectra: Magnetic vs Bending Energy Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

17 Links Energy Spectra: Magnetic vs Bending Energy Links Energy Spectrum: normalized magnetic energy vs. ropelength V = 1, Φ = 1, h = 0 tight torus: M o = ( 2π 2) 1/3 m = M λ(0) M o = ( λ 2π )4/3 Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

18 Links Energy Spectra: Magnetic vs Bending Energy Links Energy Spectrum: normalized bending energy vs. ropelength E b = 1 2 K [c(s)]2 ds tight torus: E o = πr = 2 1/3 π 5/3 ẽ = E b K = [c(s)]2 ds E o 2 4/3 π 5/3 Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

19 Links Energy Spectra: Magnetic vs Bending Energy Links Energy Spectrum: ratio between normalized magnetic energy and normalized bending energy Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

20 Vortex Knots D. Kleckner, W. Irvine (2013) Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

21 Fluid Flow Evolution Biot-Savart Law and Localized Induction Approximation (LIA) Let identify now C t with a thin vortex filament of constant circulation Γ. The vortex line moves with a self induced velocity: Biot-Savart (BS) law u(x,t) = Γ 4π ˆt (X(s) X(s )) C t X(s) X(s ) 3 ds The Biot-Savart law can be simplified according to the following: ω = ω 0ˆt ω 0 = constant R Asymptotic theory : a = δ 1 thin tube approximation X (s i) X(s j) = o(1) no self-intersection s i s j LIA law u LIA = Ẋ(s,t) X t = Γ 4π lnδ X X = Γ 4π lnδ cˆb, Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

Torus Knots and Unknots Torus Knots Theorem (Massey, 1967) A closed, non-self intersecting curve embedded in a torus Π, that cuts a meridian at p > 1 points and a longitude at q > 1 points (p and q

22 Torus Knots and Unknots Torus Knots Theorem (Massey, 1967) A closed, non-self intersecting curve embedded in a torus Π, that cuts a meridian at p > 1 points and a longitude at q > 1 points (p and q relatively prime integers), is a non-trivial knot T p,q, with winding number w = q/p. p > 1 longitudinal wraps and q > 1 meridional wraps; For given p,q (p,q coprimes) T p,q T q,p topologically equivalent. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

23 Torus Knots and Unknots Torus Unknots Case p = 1 or q = 1: Unknots U 1,m U m,1 U 0 (circle) Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

24 Torus Knots and Unknots Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

25 Torus Knots and Unknots Vortex Torus Knot Solution Under LIA Theorem (Kida, 1981) Let K v denote the embedding of a knotted vortex line in an ideal fluid. If K v evolves under LIA, then there exist a class of steady solutions in the shape of torus knots K v T p,q in terms of incomplete elliptic integrals. Solution T p,q in explicit analytic closed form, based on linear perturbation from the circular solution U 0 of NLSE: (Ricca, 1993) : r = r 0 +ǫr 1 α = s +ǫα 1 r 0 z = ˆt r 0 +ǫz 1 r = r 0 +ǫsin(wφ) α = s +ǫ 1 cos(wφ) r 0 wr 0 z = ˆt r 0 +ǫ ( 1+ 1 w 2 ) 1/2 cos(wφ) Let T p,q denote the embedding of a small-amplitude vortex torus knot K v evolving under LIA. T p,q is steady and stable under linear perturbation iff q > p (w > 1). Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

26 Torus Knots and Unknots Numerical Study of Evolution of Vortex Knots The evolution is obtained by integrating the equation of motion forward in time at each point from the initial vortex configuration (fourth order Runge Kutta algorithm); De-singularization of the BS intergral: u = u 0 +u 1 = Γ 4π (X X )ln (2l+l )1/2 Cr core +u 1 where l + and l are the distances to the nearest neighbouring point. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

27 Torus Knots and Unknots Numerical Setup Convergence tested both in space (N) and time ( t); Typical errors in computing the velocity and the energy: 10 5 cm/s and 10 7 cm 5 /s 2 respectively; Typical time step: t [0.001,0.1] chosen in relation of value N [100, 400] = number of sub-division of initial vortex configuration; The initial condition is obtained by setting: r 0 = 1, Γ = 1, ǫ 0 = 10 3, δ = /e 1/2 ; For the reference vortex ring: w = 1 and N = 313, L = 6.26 cm and N/L = 50. Computation performed on Intel(R) Core(2) Duo CPU T GHZ, RAM 1.99 GB; Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

28 Torus Knots and Unknots Translation Velocity of Vortex Knots and Unknots Velocity decreases with increasing winding number; Fastest torus knots: highest number of longitudinal wraps; At high winding number torus knots/unknots reverse their velocity. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

29 Torus Knots and Unknots Translation Velocity of Vortex Knots and Unknots Vortex knots travel faster then their corresponding unknots; The higher is the number of longitudinal wraps p, the faster is the translational motion; Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

30 Torus Knots and Unknots Kinetic Energy of Vortex Knots for w < 1 LIA law underestimates the actual energy of vortex knot; for w > 1 LIA provides much higher energy values. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

31 Torus Knots and Unknots Kinetic Energy of Vortex Unknots for w < 1 LIA law underestimates the actual energy of vortex unknot; for w > 1 LIA provides much higher energy values. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

32 Torus Knots and Unknots Structural Stability For w < 1 the space traveled tends to decrease with increasing knot complexity: the stabilizing effect due to the BS is confirmed; For w < 1 the vortex knots/unknots are LIA-unstable. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

33 Elastic Knotted Filaments Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

Knots and Links in biology Knots and Links in biology The cell produces and uses a number of types of the enzyme TOPOISOMERASE to control cellular DNA topology and geometry; DNA knots are bad for the

34 Knots and Links in biology Knots and Links in biology The cell produces and uses a number of types of the enzyme TOPOISOMERASE to control cellular DNA topology and geometry; DNA knots are bad for the cell: Knotting promotes replicon loss by blocking DNA replication; Knotting causes mutation; Knotting blocks gene transcription; Knotted DNA is potentially toxic and may drive genetic evolution; Type II topoisomerases resolve DNA knots in a single step. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

35 Coiling Under Elastic Energy Minimization Kinematics of Writhing and Coiling We explore the folding mechanism by a family of time-dependent curves X = X(ξ,t) (where t is a kinematical time) a sub-class of Fourier knots, given by X = X(ξ,t) : x = [a 1(t)cos(mξ)+a 2(t)cos(nξ)]/l(t), y = [a 3(t)sin(mξ)+a 4(t)sin(nξ)]/l(t), z = [a 5(t)sin(ξ)]/l(t). where a i(t) t = 1,...,5 are time dependent functions, m,n > 0 and length function l(t) = 1 2π 2π 0 [ ( x ) 2 + ξ ( ) 2 y + ξ ( ) ] 2 1/2 z dξ. ξ Initial condition, (t = 0): X(0,ξ) is a circle: Wr = 0; Lk = Tw; Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

36 Coiling Under Elastic Energy Minimization Bending and Mean Twist Energy Let us consider the linear elastic theory for a uniformly homogeneous and isotropic filament (χ = K b /K t = 1). The deformation energy is given (to first order) by Ẽ = Ẽb +Ẽtw +...(higher order terms) Ẽ b (t) = E b(t) = 1 (c(ξ,t)) 2 X (ξ) dξ norm. bending energy E 0 2π C Ẽ tw = E t Ω0 = (Lk Wr(t)) 2 norm. mean twist energy ; Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

37 Coiling Under Elastic Energy Minimization Coiling Under Elastic Energy Minimization The problem is modelled as follows: min a i (t) i=1,...,5 s.t. tfin t 0 Ẽ(a i(t))dt, d i = 1,...,5, t [t0,t fin], Ẽ(a i(t 0)) = Ẽ0, i = 1,...,5, l(a i(t)) = 2π, i = 1,...,5, t [t 0,t fin ], where Ẽ(a i(t)) = Ẽb(a i(t))+ẽtw(ai(t)); Initial condition: X(t 0,ξ): circle of unitary radius (Wr = 0); Ẽ(t 0 ) = 10. According to the Michell-Zajac instability: Tw c = 3 χ R we set Lk = Tw c = 3. [Maggioni, Potra, Bertocchi (2013)] Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

38 Coiling Under Elastic Energy Minimization Numerical Methods The problem is approximated by considering the following discrete version: min a i (t f ) i=1,...,5 where q Ẽ(t0)+Ẽ(tF) 2 F Ẽ(a i(t f )) +γ F 1 + f=1 f=0 F 1 p(t f )+µ f=1 s.t. Ẽ(a i(t f )) Ẽ(ai(t f+1)), i = 1,...,5, f = 0,...,F 1, Ẽ(a i(t 0)) = Ẽ0, i = 1,...,5, l(a i(t f )) = 2π, i = 1,...,5, f = 0,...,F, F: number of equidistant intervals of width q = t fin t 0 F ; p(t f ): penalization on distance of a i(t f ), i = 1,...,5 from zero (cost γ); h(t f ): penalization on curvature of a i(t f ), i = 1,...,5 (cost µ < γ). h(t f ) Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

39 Coiling Under Elastic Energy Minimization Kinematics by Cubic B-Spline Functions B-splines: Let T = {t 0,t 1,...,t m} be the knot vector; a i(t) = m n 1 w=0 P i,wb n w(t), i = 1,...,5, where P i,w, w = 0,...,m n 1 are de Boor points and b n w(t) the basis B-spline defined using the Cox-de Boor recursion formula: { b 0 1 if tw < t < t w+1 and t w < t w+1 w(t) = 0 otherwise tw+j+1 t b j w(t) = t tw b j 1 w (t)+ b j 1 t w+j t w t w+j+1 t w+1(t), w+1 j = 1,...,n. b 3 w t t 0.2 Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

40 Numerical Results Computational Framework The problem has been solved under Mathematica 8.0 environment: Global Optimization: NMinimize Nelder Mead Algorithm; Local Optimization Software: Knitro release 7 Interior Point Method; Numerical Setting: Penalization costs: µ = 0.001, γ = 0.1; [t 0,t fin ] = [0,1] is divided in F = 4 subintervals of constant width 0.25; Single and double integrals: trapezoidal rule (10 segments, 100 sub-rect.); 30 free variables (P i,w, w = 1,...,6, i = 1,...,5), 20 nonlinear inequal.; Robustness: convergence with respect to µ, γ and F has been tested; execution time: (on a pc VAIO Intel Core i5 processors, 4GB memory): seconds by means of the Nelder Mead Algorithm; under Knitro (Interior Point Algorithm). Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

41 Numerical Results Energetics and Geometric Quantities Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

42 Numerical Results Inflexional States Generic behaviour [Moffatt & Ricca, 1992]: τ (ξ,t) = X X X X X 2 as {ξ,t} {ξ i,t i}, but [T ] = 1. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

43 Numerical Results One Coil Formation Figure : One coil formation solution by means of the cubic B-spline interpolation method. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

44 Conclusions The effect of geometric and topological aspects on the dynamics and energetics of knots and links have been analyzed in different contexts: Ideal magnetohydrodynamics Euler fluids Elastic filaments Results show that: In the ideal magnetohydrodynamics context groundstate magnetic and bending energy spectra show remarkable similarities; In Euler fluids context we have: for w < 1 the more complex the vortex structure is, the faster it moves. For w > 1 all vortex structures move essentially as fast as U 0, almost independently from their total twist. The LIA law tend to under-estimate the energy of knots with w < 1 and to over-estimate the energy of knots with w > 1. The stabilizing effect of the Biot-Savart law for knots with w < 1 (LIA-unstable) has been confirmed. Kinematics for Reidemeiste type I move as solution of the elastic energy minimization problem are proposed. Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

45 References References Scharein, R.G. Cantarella, J., Rawdon, E., Piatek M. Kleckner, D., Irvine, W. (2013) Creation and dynamics of knotted vortices, Nature Physics 9, Maggioni, F., Ricca, R.L. (2009) On the groundstate energy of tight knots, Proc. R. Soc. A. 465, Maggioni, F., Alamri S., Barenghi C.F., Ricca R.L. (2010) Velocity, energy and helicity of vortex knots and unknots, Phys. Rev. E, 82(02639), 1 9. Maggioni, F., Potra, F. A., Bertocchi, M. (2013) Optimal kinematics of a looped filament, J. Optim. Theory Appl., 159, Ricca, R.L., Maggioni, F. (2014) The energy spectrum of knots and links, Journal of Physics A: Mathematical and Theoretical, 47(20), Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

References Knots in Hellas 17-23 July 2016 Francesca

46 References Knots in Hellas July 2016 Francesca Maggioni L energia minima dei nodi Brescia, 17/05/ / 46

Velocity, Energy and Helicity of Vortex Knots and Unknots. F. Maggioni ( )

Velocity, Energy and Helicity of Vortex Knots and Unknots F. Maggioni ( ) Dept. of Mathematics, Statistics, Computer Science and Applications University of Bergamo (ITALY) ( ) joint work with S. Alamri,