Weaving Knotted Vector Fields with Tunable Helicity

Abstract

General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms We present a general construction of divergence-free knotted vector fields from complex scalar fields, whose closed field lines encode many kinds of knots and links, including torus knots, their cables, the figure-8 knot, and its generalizations. As finite-energy physical fields, they represent initial states for fields such as the magnetic field in a plasma, or the vorticity field in a fluid. We give a systematic procedure for calculating the vector potential, starting from complex scalar functions with knotted zero filaments, thus enabling an explicit computation of the helicity of these knotted fields. The construction can be used to generate isolated knotted flux tubes, filled by knots encoded in the lines of the vector field. Lastly, we give examples of manifestly knotted vector fields with vanishing helicity. Our results provide building blocks for analytical models and simulations alike. DOI: 10.1103/PhysRevLett.117.274501 Introduction.-The idea that a physical field-such as a magnetic field-could be weaved into a knotty texture has fascinated scientists ever since Lord Kelvin conjectured that atoms were, in fact, vortex knots in the aether. Since then, topology has emerged as a key organizing principle in physics, and knottiness is being explored as a fundamental aspect of classical and quantum fluids [1][2][3][4][5][6][7][8], magnetic fields in light and plasmas [9][10][11][12][13][14][15][16][17][18][19], liquid crystals [20][21][22][23], optical fields [24,25], nonlinear field theories [26][27][28][29], wave chaos [30], and superconductors [31,32].In particular, helicity-a measure of average linking of field lines-is a conserved quantity in ideal fluids [33,34] and plasmas [35][36][37]. Helicity thus places a fundamental topological constraint on their evolution [1,10], and plays an important role in turbulent dynamo theory [38][39][40], magnetic relaxation in plasmas [41][42][43], and turbulence [44,45]. Beyond fluids and plasmas, helicity conservation leads to a natural connection between the minimum energy configurations of knotted magnetic flux tubes [10,42,46] and tight knot configurations [47,48], and tentatively with the spectrum of mass energies of glueballs in the quarkgluon plasma [49][50][51].Knotted field configurations provide a natural setting for studying helicity, but more subtlety is required to tie a knot in the lines of a vector field than in a shoelace: all the streamlines of the entire space-filling field must twist to conform to the knotted region. The difficulty of constructing knotted field configurations with controlled helicity makes it challenging to understand the role of helicity in the evolution of knotted structures [1,10,12].In this Letter, we show how to explicitly construct knotted, divergence-free vector fields with a wide range of topolo...

When do knots in light stay knotted?

“…The results presented above could be used in conjunction with methods for the construction of knotted vector fields [35] to design null electromagnetic fields with nontrivial topology.…”

“…A knotted divergence-free vector field can be constructed from a rational map ψ(r) based on Milnor polynomials [5], as shown in [35][36][37]:…”

Spin-Orbital Momentum Decomposition and Helicity Exchange in a Set of Non-Null Knotted Electromagnetic Fields

“…The relation between the helicity basis and the planar Fourier basis can be obtained by comparing Equations (24) and (26). Consequently, the electric and magnetic fields of an electromagnetic field in a vacuum, and the vector potentials in the Coulomb gauge can be expressed in this basis as:…”

“…In this paper, we first make a brief review of the concept of electromagnetic duality. That duality, termed "electromagnetic democracy" [10], has been central in the work of knotted field configurations [5,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Related field configurations have also appeared in plasma physics [27][28][29][30], optics [31][32][33][34][35], classical field theory [36], quantum physics [37,38], various states of matter [39][40][41][42][43] and twistors [44,45].…”

Knotted optical vortices in exact solutions to Maxwell's equations

“…These solutions were shown to be equivalent to elementary states arising in twistor theory [7], which allowed for a generalization of the knotted solutions to other massless field equations, in particular the linearized Einstein equations [12]. More recently Kedia et al [13] have proposed a method capable of constructing divergence-free vector fields with knotted field lines more general than torus knots. However, a knotted structure can also be encoded in optical vortices, the lines of zero intensity of an electromagnetic field.…”