We construct analytically, a new family of null solutions to Maxwell's equations in free space whose field lines encode all torus knots and links. The evolution of these null fields, analogous to a compressible flow along the Poynting vector that is shear free, preserves the topology of the knots and links. Our approach combines the construction of null fields with complex polynomials on S3. We examine and illustrate the geometry and evolution of the solutions, making manifest the structure of nested knotted tori filled by the field lines.

Helicity, a topological measure of the intertwining of vortices in a fluid flow, is a conserved quantity in inviscid fluids but can be dissipated by viscosity in real flows. Despite its relevance across a range of flows, helicity in real fluids remains poorly understood because the entire quantity is challenging to measure. We measured the total helicity of thin-core vortex tubes in water. For helical vortices that are stretched or compressed by a second vortex, we found conservation of total helicity. For an isolated helical vortex, we observed evolution toward and maintenance of a constant helicity state after the dissipation of twist helicity by viscosity. Our results show that helicity can remain constant even in a viscous fluid and provide an improved basis for understanding and manipulating helicity in real flows.

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...

In addition to mass, energy, and momentum, classical dissipationless flows conserve helicity, a measure of the topology of the flow. Helicity has far-reaching consequences for classical flows from Newtonian fluids to plasmas. Since superfluids flow without dissipation, a fundamental question is whether such a conserved quantity exists for superfluid flows. We address the existence of a "superfluid helicity" using an analytical approach based on the symmetry underlying classical helicity conservation: the particle relabeling symmetry. Furthermore, we use numerical simulations to study whether bundles of superfluid vortices which approximate the structure of a classical vortex, recover the conservation of classical helicity and find dynamics consistent with classical vortices in a viscous fluid. 1 From here on, we refer to classical inviscid isentropic flows as Euler flows 2 We shall only consider superfluids with a complex scalar order parameter as in 4 He and atomic Bose-Einstein condensates.arXiv:1708.01526v4 [cond-mat.quant-gas]