
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 S^{3}. We examine and illustrate the geometry and evolution of the solutions, making manifest the structure of nested knotted tori filled by the field lines.
Knots and the application of mathematical knot theory to spacefilling fields are enriching our understanding of a variety of physical phenomena with examples in fluid dynamics [1–3], statistical mechanics [4], and quantum field theory [5], to cite a few. Knotted structures embedded in physical fields, previously only imagined in theoretical proposals such as Lord Kelvin’s vortex atom hypothesis [6], have in recent years become experimentally accessible in a variety of physical systems, for example, in the vortex lines of a fluid [7–9], the topological defect lines in liquid crystals [10,11], singular lines of optical fields [12], magnetic field lines in electromagnetic fields [13–15], and in spinor BoseEinstein condensates [16]. Furthermore, numerical simulations have shown that stable knotlike structures arise in the SkyrmeFaddeev model [17,18], and consequently in triplet superconductors [19,20] and charged Bose condensates [21]. Analytical solutions for such excitations, however, are difficult to construct owing to the inherent nonlinearity in most dynamical fields and have therefore remained elusive.
An exception is a particularly elegant solution to Maxwell’s equations in free space (see Fig. 1), brought to light by Raсada [22], which provides an encouraging manifestation of a persistent nontrivial topological structure in a linear field theory. This solution, referred to as the Hopfion solution for the rest of the Letter, can furthermore be experimentally realized using tightly focused LaguerreGaussian beams [14].
In this Letter, we present the first example of a family of exact knotted solutions to Maxwell’s equations in free space, with the electric and magnetic field lines encoding all torus knots and links, which persist for all time. The unique combination of experimental potential and opportunity for analytical study makes light an ideal candidate for studying knotted field configurations and furthermore, a means of potentially transferring knottedness to matter.
In the case of the Hopfion solution illustrated in Fig. 1, the electric, magnetic, and Poynting field lines exhibit a remarkable structure known as a Hopf fibration, with each field line forming a closed loop such that any two loops are linked. At time t = 0, each of the electric, magnetic, and Poynting field lines have identical structure (that of a Hopf fibration), oriented in space so that they are mutually orthogonal to each other. The topology of these structures is preserved with time, as the electric and magnetic field lines evolve like unbreakable filaments embedded in a fluid flow, stretching and deforming while retaining their identity [15,23]. The Poynting field lines evolve instead via a rigid translation along the z axis. The Hopfion solution has been rediscovered and studied in several contexts [14,22,24–27] and can be constructed in many ways using complex scalar maps, spinors, twistors.
Despite numerous attempts at generalizing the Hopfion solution to light fields encoding more complex knots, the problem of constructing light fields encoding knots that are preserved in time has remained open until now. Attempts at generalizing Hopfions to torus knots [14,15,28] succeeded at constructing such solutions at an instant in time, but their structure was not preserved [15], and unraveled with time. Beyond Maxwell’s equations, the more general problem of finding explicit solutions to dynamical flows which embody persistent knots has also remained open.
The fluidlike topologypreserving evolution of the Hopfion solution is closely tied to the property that the electric and magnetic fields are everywhere perpendicular and of equal magnitude [a constraint known as nullness, cf. Eq. (3)]. Nullness introduces an effective nonlinearity in the problem and imposes a dynamical geometric constraint on Maxwell fields, restricting the space of possible topological configurations of field lines.
We construct knotted solutions within the space of null field configurations by making use of formalisms developed for the construction of null Maxwell fields, such as Bateman’s method [29] or equivalently a spinor formalism (see Supplemental Material [30]). The combination of a null electromagnetic field formalism with a topological construction, leading to a family of knotted null solutions is the central result of this Letter.We now briefly review the key features of the evolution of null electromagnetic fields.
Null electromagnetic fields. — Null electromagnetic fields have a rich history, from the early construction by Bateman [29] to Robinson’s theorem [31] and Penrose’s twistor theory [32]. For a null electromagnetic field, the Poynting field not only guides the flow of energy, but also governs the evolution of the electric and magnetic field lines. These field lines evolve as though embedded in a fluid, flowing at the speed of light, in the direction of the Poynting field [15,23]. The persistence of the null conditions guarantees the continued fluidlike evolution of the electric and magnetic field lines, giving them the appearance of unbreakable elastic filaments.
FIG. 1 (color online). Hopfion solution: field line structure (a)–(c) and time evolution (d)–(e). Field lines fill nested tori, forming closed loops linked with every other loop. (a) Hopf link formed by the circle at the core (orange) of the nested tori, and one of the field lines (blue). (b) The torus (purple) that the field line forming the Hopf link is tangent to. (c) Nested tori (purple) enclosing the core, on which the field lines lie. (d) Time evolution of the Poynting field lines (gray), an energy isosurface (red), and the energy density (shown via projections). (e) Time evolution of the electric (yellow), magnetic (blue), and Poynting field lines (gray), with the top view shown in the inset.