aa r X i v : . [ m a t h - ph ] F e b STABLE KNOTS AND LINKS IN ELECTROMAGNETIC FIELDS
BENJAMIN BODEA
BSTRACT . In null electromagnetic fields the electric and the magnetic field lines evolvelike unbreakable elastic filaments in a fluid flow. In particular, their topology is preservedfor all time. We prove that for every link L there is such an electromagnetic field thatsatisfies Maxwell’s equations in free space and that has closed electric and magnetic fieldlines in the shape of L for all time.
1. I
NTRODUCTION
Knotted structures appear in physical fields in a wide range of areas of theoreticalphysics; in liquid crystals [17, 22, 23], optical fields [8], Bose-Einstein condensates [25],fluid flows [11, 12], the Skyrme-Faddeev model [30] and several others.Mathematical constructions of initially knotted configurations in physical fields makeexperiments and numerical simulations possible. However, the knot typically changes ordisappears as the field evolves with time as prescribed by some differential equation orenergy functional. There are some results regarding the existence of stationary solutionsof the harmonic oscillator and the hydrogen atom [9, 10], and the existence of solutions tocertain Schr¨odinger equations that describe any prescribed time evolution of a knot [13].In particular, this implies the existence of solutions that contain a given knot for all time,i.e., the knot is stable or robust . However, more general (i.e., regarding more generaldifferential equations) explicit analytic constructions of such solutions are not known.In the case of electromagnetic fields and Maxwell’s equations, the first knotted solutionwas found by Ra˜nada [26]. His field contains closed magnetic and electric field lines thatform the Hopf link for all time. Using methods from [4] and [20] we can algorithmicallyconstruct for any given link L a vector field B : R → R that has a set of closed field linesin the shape of L and that can be taken as an initial configuration of the magnetic partof an electromanetic field, say at time t =
0. However, these links cannot be expected tobe stable, since they usually undergo reconnection events as time progresses and the fieldevolves according to Maxwell’s equations, or they disappear altogether. Necessary andsufficient conditions for the stability of knotted field lines are known [21], but so far onlythe family of torus links has been constructed and thereby been proven to arise as stableknotted field lines in electromagnetism.In [19] Kedia et al. offer a construction of null electromagnetic fields with stable toruslinks as closed electric and magnetic field lines using an approach developed by Bateman[3]. In this article we prove that their construction can be extended to any link type, imply-ing the following result:
Theorem 1.1.
For every n-component link L = L ∪ L ∪ · · · ∪ L n and every subset I ⊂{ , , . . ., n } there is an electromagnetic field F that satisfies Maxwell’s equations in freespace and that has a set of closed field lines (electric or magnetic) ambient isotopic to Lfor all time, with closed electric field lines that are ambient isotopic to S i ∈ I L i for all timeand closed magnetic field lines that are ambient isotopic to S i / ∈ I L i for all time. This shows not only that every pair of links L and L can arise as a set of robustclosed electric and magnetic field lines, respectively, but also that any linking between thecomponents of L and L can be realised.We would like to point out that the subset I of the set of components of L does notneed to be non-empty or proper for the theorem to hold. As a special case, we may choose L and I such that S i ∈ I L i and S i / ∈ I L i are ambient isotopic, which shows the followinggeneralisation of the results in [19]. Corollary 1.2.
For any link L there is an electromagnetic field F that satisfies Maxwell’sequations in free space and whose electric and magnetic field both have a set of closedfield lines ambient isotopic to L for all time. The proof of the theorem relies on the existence of certain holomorphic functions,whose explicit construction eludes us at this moment. As a consequence, Theorem 1.1guarantees the existence of the knotted fields, but does not allow us to provide any newexamples beyond the torus link family.The closed field lines at time t = R of real analyticLegendrian links with respect to the standard contact structure in S . This family of linkshas been studied by Rudolph in the context of holomorphic functions as totally tangential C -links [28, 29].The remainder of the article is structured as follows. In Section 2 we review some keymathematical concepts, in particular Bateman’s construction of null electromagnetic fieldsand knots and their role in contact geometry. Section 3 summarises some observationsthat relate the problem of constructing knotted field lines to a problem on holomorphicextendability of certain functions. The proof of Theorem 1.1 can be found in Section 4,where we use results by Rudolph, Burns and Stout to show that the functions in questioncan in fact be extended to holomorphic functions. In Section 5 we offer a brief discussionof our result and some properties of the resulting electromagnetic fields. Acknowledgements:
The author is grateful to Mark Dennis, Daniel Peralta-Salas andVera Vertesi for helpful discussions. The author was supported by JSPS KAKENHI GrantNumber JP18F18751 and a JSPS Postdoctoral Fellowship as JSPS International ResearchFellow. 2. M
ATHEMATICAL BACKGROUND
Knots and links.
For m ∈ N we write S m − for the ( m − ) -sphere of unit radius:(1) S m − = { ( z , z , . . . , z m ) ∈ C : m ∑ i = | z i | = } . Via stereographic projection we have S ∼ = R ∪ { ∞ } . A link with n components in a 3-manifold M is (the image of) a smooth embedding of n circles S ⊔ S ⊔ . . . ⊔ S in M . Alink with only one component is called a knot . The only 3-manifolds that are relevant forthis article are M = S and M = R .Knots and links are studied up to ambient isotopy or, equivalently, smooth isotopy, thatis, two links are considered equivalent if one can be smoothly deformed into the otherwithout any cutting or gluing. This defines an equivalence relation on the set of all linksand we refer to the equivalence class of a link L as its link type or, in the case of a knot, as its knot type . It is very common to be somewhat lax with the distinction between the concept TABLE KNOTS AND LINKS IN ELECTROMAGNETIC FIELDS 3 of a link and its link type. When there is no risk of confusion we will for example refer to a link L even though we really mean the link type, i.e., the equivalence class, representedby L .One special family of links/link types is the family of torus links T p , q and the equiva-lence classes that they represent. It consists of all links that can be drawn on the surfaceof an unknotted torus T = S × S in R or S and they are characterised by two integers p and q , the number of times the link winds around each S . This definition leaves an ambi-guity regarding the sign of p and q , i.e., which direction is considered as positive wrappingaround the meridian and the longitude. This ambiguity is removed by the standard conven-tion to choose(2) ( ρ e i q ϕ , q − ρ e i p ϕ ) as a parametrisation of the ( p , q ) -torus knot in the unit 3-sphere S ⊂ C with p , q > ϕ ranges from 0 to 2 π and ρ is the solution to ρ | p | = p − ρ | q | . Itfollows that for positive p and q the complex curve z p − z q = S in the ( p , q ) -torus knot T p , q [24].Knot theory is now a vast and quickly developing area of mathematics with many con-nections to biology, chemistry and physics. For a more extensive introduction we refer theinterested reader to the standard references [1, 27]. The role that knots play in physics isdiscussed in more detail in [2, 18].2.2. Bateman’s construction.
Our exposition of Bateman’s work follows the relevantsections in [19]. In electromagnetic fields that are null for all time the electric and mag-netic field lines evolve like unbreakable elastic in an ideal fluid flow. They are draggedin the direction of the Poynting vector field with the speed of light [16, 21]. This meansthat the link types of any closed field lines remain unchanged for all time. In the follow-ing we represent a time-dependent electromagnetic field by its Riemann-Silberstein vector F = E + i B , where E and B are time-dependent real vector fields on R , representing theelectric and magnetic part of F , respectively.It was shown in [21] that the nullness condition(3) E · B = , E · E − B · B = , for all t ∈ R is equivalent to F being both null and shear-free at t =
0, that is,(4) ( E · B ) | t = = , ( E · E − B · B ) | t = = , and (( E i E j − B i B j ) ∂ j V i ) | t = = , (( E i B j + E j B i ) ∂ j V i ) | t = = , (5)where V = E × B / | E × B | is the normalised Poynting field and the indices i , j = , , E = ( E , E , E ) , B = ( B , B , B ) and V = ( V , V , V ) .It is worth pointing out that the Poynting vector field V of a null field satisfies the Eulerequation for a pressure-less flow:(6) ∂ t V + ( V · ∇ ) V = . More analogies between null light fields and pressure-less Euler flows are summarised in[21].The transport of field lines by the Poynting field of a null electromagnetic field wasmade precise in [21]. We write W = ( E · E + B · B ) for the electromagnetic density. The BENJAMIN BODE normalised Poynting vector field V transports (where it is defined) E / W and B / W . In thefollowing construction V can be defined everywhere and since, ∂ t W + ∇ · ( W V ) =
0, thenodal set of W is also transported by V . This implies that if L is a link formed by closedelectric field lines at time t = L is a link formed by closed magnetic field lines ofsuch an electromagnetic field at t = W = L and L ), then theirtime evolution according to Maxwell’s equations does not only preserve the link types of L and L , but also the way in which they are linked, i.e., the link type of L ∪ L .Bateman discovered a construction of null electromagnetic fields [3], which guaranteesthe stability of links and goes as follows. Take two functions α , β : R × R → C that satisfy(7) ∇α × ∇β = i ( ∂ t α∇β − ∂ t β∇α ) , where ∇ denotes the gradient with respect to the three spatial variables.Then for any pair of holomorphic functions f , g : C → C the field defined by(8) F = E + i B = ∇ f ( α , β ) × ∇ g ( α , β ) satisfies Maxwell’s equations and is null for all time. The field F can be rewritten as(9) F = h ( α , β ) ∇α × ∇β , where h = ∂ z f ∂ z g − ∂ z f ∂ z g and ( z , z ) are the coordinates in C . Since f and g arearbitrary holomorphic functions, we obtain a null field for any holomorphic function h : C → C .Kedia et al. used Bateman’s construction to find concrete examples of electromagneticfields with knotted electric and magnetic field lines [19]. In their work both the electricand the magnetic field lines take the shape of torus knots and links. They consider α = x + y + z − t − + zx + y + z − ( t − i ) , β = ( x − i y ) x + y + z − ( t − i ) , (10)where x , y and z are the three spatial coordinates and t represents time. It is a straight-forward calculation to check that α and β satisfy Equation (7). Note that for any valueof t = t ∗ , the function ( α , β ) | t = t ∗ : R → C gives a diffeomorphism from R ∪ { ∞ } to S ⊂ C .The construction of stable knots and links in electromagnetic fields therefore comesdown to finding holomorphic functions f and g , or equivalently one holomorphic function h . Since the image of ( α , β ) is S , it is not necessary for these functions to be holomorphic(or even defined) on all of C . It suffices to find functions that are holomorphic on an openneighbourhood of S in C .Kedia et al. find that for f ( z , z ) = z p and g ( z , z ) = z q the resulting electric andmagnetic fields both contain field lines that form the ( p , q ) -torus link T p , q . Hence there isa construction of flow lines in the shape of torus links that are stable for all time. Remark 2.1.
It was wrongly stated in [19] and [5] that for t = the map ( α , β ) in Equation(10) is the inverse of the standard stereographic projection. In fact, the inverse of thestandard stereographic projection is given by ( u , v ) : R → S ,u = x + y + z − + zx + y + z + , v = ( x + i y ) x + y + z + , (11) TABLE KNOTS AND LINKS IN ELECTROMAGNETIC FIELDS 5 so that ( α , β ) | t = is actually the inverse of the standard stereographic projection followedby a mirror reflection that sends Im ( z ) to − Im ( z ) or equivalently it is a mirror reflec-tion in R along the y = -plane followed by the inverse of the standard stereographicprojection.Kedia et al.’s choice of f and g was (in their own words) ‘guided’ by the hypersurfacez p ± z q = . Complex hypersurfaces like this and their singularities have been extensivelystudied by Milnor and others [6, 24] and it is well-known that the hypersurface intersectsS in the ( p , q ) -torus knot T p , q . Even though this made the choice of f and g somewhatintuitive (at least for Kedia et al.), there seems to be no obvious relation between the hyper-surface and the electromagnetic field that would enable us to generalise their approach.Since their fields contain the links T p , q in R , the corresponding curves on S are actuallythe mirror image T p , − q . Therefore, it seems more plausible that (if there is a connectionto complex hypersurfaces at all) the relevant complex curve is z p z q − = , which inter-sects a 3-sphere of an appropriate radius in T p , − q [28] . However, in contrast to Milnor’shypersurfaces, this intersection is totally tangential, i.e., at every point of intersection thetangent plane of the hypersurface lies in the tangent space of the 3-sphere. This is an in-teresting property that plays an important role in the generalisation of the construction toarbitrarily complex link types in the following sections. Contact structures and Legendrian links. A contact structure on a 3-manifold M is a smooth, completely non-integrable plane distribution ξ ⊂ T M in the tangent bundle of M . It can be given as the kernel of a differential 1-form, a contact form α , for which thenon-integrability condition reads(12) α ∧ d α = . It is a convention to denote contact forms by α . This should not be confused with the firstcomponent of the map ( α , β ) in Equation (10). Within this subsection α refers to a contactform, in all other sections it refers to Equation (10). The choice of α for a given ξ is notunique, but the non-integrability property is independent of this choice.In other words, for every point p ∈ M we have a plane (a 2-dimensional linear subspace) ξ p in the tangent space T p ( M ) given by ξ p = ker p α , which is the kernel of α when α isregarded as a map T p M → R . The non-integrability condition ensures that there is a certaintwisting of these planes throughout M . We call the pair of manifold M and contact structure ξ a contact manifold ( M , ξ ) .The standard contact structure ξ on S is given by the contact form(13) α = ∑ j = ( x j d y j − y j d x j ) , where we write the complex coordinates ( z , z ) of C in terms of their real and imaginaryparts: z j = x j + i y j .There are two interesting geometric interpretations of the standard contact structure ξ .Firstly, the planes are precisely the normals to the fibers of the Hopf fibration S → S .Secondly, the planes are precisely the complex tangent lines to S .A link L in a contact manifold ( M , ξ ) is called a Legendrian link with respect to thecontact structure ξ , if it is everywhere tangent to the contact planes, i.e., T p L ⊂ ξ p . It isknown that every link type in S has representatives that are Legendrian. In other words, forevery link L in S there is a Legendrian link with respect to the standard contact structureon S that is ambient isotopic to L . BENJAMIN BODE
More details on contact geometry and the connection to knot theory can be found in[14, 15]. 3. L
EGENDRIAN FIELD LINES
In this section we would like to point out some observations on Bateman’s construction.Bateman’s construction turns the problem of constructing null fields with knotted field linesinto a problem of finding appropriate holomorphic functions h : C → C . Our observationsturn this into the question whether for a given Legendrian link L with respect to the standardcontact structure on S a certain function defined on L admits a holomorphic extension. Lemma 3.1.
Let h : C → C be a function that is holomorphic on an open neighbourhoodof S and let F = h ( α , β ) ∇α × ∇β be the corresponding electromagnetic field with ( α , β ) as in Equation (10). Suppose L is a set of closed magnetic field lines or a set of closedelectric field lines of F at time t = . Then ( α , β ) | t = ( L ) is a Legendrian link with respectto the standard contact structure on S .Proof : It is known that all fields that are constructed with the same choice of ( α , β ) have the same Poynting field, independent of h . For ( α , β ) as in Equation (10) with t = ( α , β ) | t = is tangent to the fibers of the Hopf fibration. By the definitionof the Poynting field, the electric and magnetic field are orthogonal to the Poynting fieldand it is a simple calculation that their pushforwards by ( α , β ) | t = are orthogonal as well.Therefore, they must be normal to the fibers of the Hopf fibration. Hence the pushforwardof all electric and magnetic field lines by ( α , β ) are tangent to the standard contact structureon S . In particular, any closed electric or magnetic field line is a Legendrian link withrespect to the standard contact structure. (cid:3) A more general statement of Lemma 3.1 is proven in [5]. It turns out that ( α , β ) define acontact structure for each value of t , where time evolution is given by a 1-parameter familyof contactomorphisms, and all sets of closed flow lines at a fixed moment in time are (theimages in R of) Legendrian links with respect to the corresponding contact structure.Lemma 3.1 tells us that (the projection of) closed field lines form Legendrian links. Wewould like to go in the other direction, starting with a Legendrian link and constructing acorresponding electromagnetic field for it.We define the map ϕ = ( α , β ) | t = : R ∪ { ∞ } → S . The particular choice of ( α , β ) in Equation (10) does not only determine a contact structure, but also provides us with anexplicit orthonormal basis of the plane ξ p in T p S for all p ∈ S \{ ( , ) } , given by(14) ξ p = span { v , v } where v and v are given by v = ϕ ∗ (cid:18) ( x + y + z + ) (cid:0) ∇α (cid:12)(cid:12) t = × ∇β (cid:12)(cid:12) t = (cid:1)(cid:19) = − x ∂∂ x + y ∂∂ y + x ∂∂ x − y ∂∂ y , v = ϕ ∗ (cid:18) ( x + y + z + ) (cid:0) ∇α (cid:12)(cid:12) t = × ∇β (cid:12)(cid:12) t = (cid:1)(cid:19) = − y ∂∂ x − x ∂∂ y + y ∂∂ x + x ∂∂ y . (15) TABLE KNOTS AND LINKS IN ELECTROMAGNETIC FIELDS 7
They are pushforwards of multiples of Re ( ∇α × ∇β ) | t = and Im ( ∇α × ∇β ) | t = by ϕ . It iseasy to see from these expressions that v and v are orthonormal and span the contact plane ξ p at each point p ∈ S \{ ( , ) } . The point p = ( , ) is excluded, since it is ( , ) = ϕ ( ∞ ) .A magnetic field B constructed using Bateman’s method satisfies B = Im ( F )= Re ( h ( α , β )) Im ( ∇α × ∇β ) + Im ( h ( α , β )) Re ( ∇α × ∇β ) , (16)while the electric field E satisfies E = Re ( F )= Re ( h ( α , β )) Re ( ∇α × ∇β ) − Im ( h ( α , β )) Im ( ∇α × ∇β ) . (17)In particular, both fields are at every point a linear combination of Re ( ∇α × ∇β ) andIm ( ∇α × ∇β ) and their pushforwards by ϕ are linear combinations of v and v . Thefact that v and v are a basis for the contact plane ξ p for all p ∈ S \{ ( , ) } impliesthat Equations (16) and (17) provide an alternative proof of Lemma 3.1. Hence everyclosed field line must be a Legendrian knot and the holomorphic function h describes thecoordinates of the field with respect to this preferred basis.Suppose now that we have an n -component Legendrian link L = L ∪ L ∪ . . . ∪ L n withrespect to the standard contact structure on S , with ( , ) L , a subset I ⊂ { , . . . , n } , anda non-zero section X of its tangent bundle T L ⊂ ξ ⊂ T S . We can define a complex-valuedfunction H : L → C given byRe ( H ( z , z )) = X ( z , z ) · v , for all ( z , z ) ∈ L i , i ∈ I Im ( H ( z , z )) = − X ( z , z ) · v , for all ( z , z ) ∈ L i , i ∈ I , Re ( H ( z , z )) = X ( z , z ) · v , for all ( z , z ) ∈ L i , i / ∈ I Im ( H ( z , z )) = X ( z , z ) · v , for all ( z , z ) ∈ L i , i / ∈ I , (18)where · denotes the standard scalar product in R = T ( z , z ) C . Proposition 3.2.
If there is an open neighbourhood U of S ⊂ C and a holomorphic func-tion h : U → C with h | L = H, then the corresponding electromagnetic field F = h ( α , β ) ∇α × ∇β at t = has closed field lines ambient isotopic to (the mirror image of) L, with closedelectric field lines in the shape of (the mirror image of) S i ∈ I L i and magnetic field lines inthe shape of (the mirror image of) S i / ∈ I L i .Proof : For every point q ∈ ϕ − ( S i / ∈ I L i ) we have B | t = ( q ) = ( Re ( h ( α , β )) Im ( ∇α × ∇β )+ Im ( h ( α , β )) Re ( ∇α × ∇β )) (cid:12)(cid:12)(cid:12) t = , ( x , y , z )= q = ( | q | + ) (cid:0) (( Re ( H ( α , β ))( ϕ − ) ∗ ( v )+ Im ( H ( α , β ))( ϕ − ) ∗ ( v )) (cid:1)(cid:12)(cid:12)(cid:12) t = , ( x , y , z )= q = ( | q | + ) ( ϕ − ) ∗ ( X ( α , β ) ) (cid:12)(cid:12)(cid:12) t = , ( x , y , z )= q , (19)where | · | denotes the Euclidean norm in R . The second equality follows from h | L = H and Equation (15). The last equality follows from the orthonormality of the basis { v , v } ,the definition of H and the fact that L is Legendrian. Equation (19) states that at t = BENJAMIN BODE field B is everywhere tangent to ϕ − ( S i / ∈ I L i ) . In particular, at t = B has a set ofclosed flow lines that is ambient isotopic to the mirror image of S i / ∈ I L i (cf. Remark 2.1).Similarly, for every q ∈ ϕ − ( S i ∈ I L i ) we have E | t = ( q ) = ( Re ( h ( α , β )) Re ( ∇α × ∇β ) − Im ( h ( α , β )) Im ( ∇α × ∇β )) (cid:12)(cid:12)(cid:12) t = , ( x , y , z )= q = ( | q | + ) (cid:0) (( Re ( H ( α , β ))( ϕ − ) ∗ ( v ) − Im ( H ( α , β ))( ϕ − ) ∗ ( v )) (cid:1)(cid:12)(cid:12)(cid:12) t = , ( x , y , z )= q = ( | q | + ) ( ϕ − ) ∗ ( X ( α , β ) ) (cid:12)(cid:12)(cid:12) t = , ( x , y , z )= q . (20)The same arguments as above imply that at t = E is everywhere tangent to ϕ − ( S i ∈ I L i ) , so that at t = E has a set of closed flow lines that is ambientisotopic to S i ∈ I L i . (cid:3) Since the constructed fields are null for all time, the topology of the electric and mag-netic field lines does not change, and the fields contain L for all time. We hence have thefollowing corollary. Corollary 3.3.
Let L = L ∪ L ∪ . . . ∪ L n be an n-component Legendrian link with respectto the contact structure in S with I ⊂ { , . . ., n } and a non-vanishing section of its tangentbundle such that the corresponding function H : L → C allows a holomorphic extensionh : U → C to an open neighbourhood U of S . Then F = h ( α , β ) ∇α × ∇β has a set ofclosed field lines that is ambient isotopic to the mirror image of L for all time, with a setof closed electric field lines that is ambient isotopic to the mirror image of S i ∈ I L i for alltime and a set of closed magnetic field lines that is ambient isotopic to the mirror image of S i / ∈ I L i for all time. Therefore, what we have to show in order to prove Theorem 1.1 is that every link type(with every choice of a subset of its components) has a Legendrian representative as in thecorollary. 4. T
HE PROOF OF THE THEOREM
We have seen in the previous section that Theorem 1.1 can be proven by showing thatevery link type has a Legendrian representative for which a certain function has a holomor-phic extension. Questions like this, regarding the existence of holomorphic extensions offunctions defined on a subset of C m , are important in the study of complex analysis in m variables and are in general much more challenging when m >
1. In this section, we firstprove that every link type has a Legendrian representative with certain properties regardingreal analyticity. We then review a result from complex analysis by Burns and Stout thatguarantees that for this class of real analytic submanifolds of C contained in S the desiredholomorphic extension exists, thereby proving Theorem 1.1. Lemma 4.1.
Every link type has a real analytic Legendrian representative L that admits anon-zero section of its tangent bundle, such that for any given subset I of its set of compo-nents the corresponding function H : L → C as in Equation (18) is real analytic.Proof : The lemma is essentially proved in [29], where it is shown that every link has aLegendrian representative L (with respect to the contact structure in S ) that is the image TABLE KNOTS AND LINKS IN ELECTROMAGNETIC FIELDS 9 of a smooth embedding, given by a Laurent polynomial η i = ( η i , , η i , ) : S → S ⊂ C ine i χ for each component L i . The set of functions η i in [29] is obtained by approximatingsome smooth embedding, whose image is a Legendrian link L ′ of the same link type as L .It is a basic exercise in contact topology to show that we can assume that ( , ) L ′ [14]and hence also ( , ) L .Since each η i is a real analytic embedding, the inverse η − i : L → S is real analyticin x , y , x and y for all i = , , . . . , n . Likewise ∂ χ η i : S → T L i is real analytic in χ and non-vanishing, since η i is an embedding. It follows that the composition X : =( ∂ χ η i ) ◦ η − i : L i → T L i is a real analytic non-vanishing section of the tangent bundle of L i for all i = , , . . . , n . Equations (18) and (15) then directly imply that H is also realanalytic, no matter which subset I of the components of L is chosen. (cid:3) It was shown in [28] that a link L in S is a real analytic Legendrian link if and only if itis a totally tangential C -link , i.e., L arises as the intersection of a complex plane curve and S that is tangential at every point. Recall from Remark 2.1 that the torus links constructedin [19] arise in this way, where the complex plane curve is z p z q − = transverse C -links or, equivalently, quasipositivelinks , have been studied as stable vortex knots in null electromagnetic fields in [5].Following Burns and Stout [7] we call a real analytic submanifold Σ of C that is con-tained in S an analytic interpolation manifold (relative to the 4-ball B) if every real an-alytic function Σ → C is the restriction to Σ of a function that is holomorphic on someneighbourhood of B . The neighbourhood depends on the function in question. Theorem 4.2 (Burns-Stout [7]) . Σ is an analytic interpolation manifold if and only ifT p ( Σ ) ⊂ T C p ( S ) for every p ∈ Σ , where T C p ( S ) denotes the maximal complex subspace ofT p ( S ) . The result stated in [7] holds in fact for more general ambient spaces and their bound-aries, namely strictly pseudo-convex domains with smooth boundaries. The open 4-ball B with boundary ∂ B = S is easily seen to be an example of such a domain. Proof of Theorem 1.1 : By Lemma 4.1 every link type can be represented by a realanalytic Legendrian link L . It is thus a real analytic submanifold of C that is contained in S . The condition T p L ⊂ T C p ( S ) is equivalent to L being a Legendrian link with respect tothe standard contact structure on S . Hence L is an analytic interpolation manifold. SinceLemma 4.1 also implies that for every choice of I the function H : L → C can be taken tobe real analytic, Theorem 4.2 implies that H is the restriction of a holomorphic function h : U → C , where U is some neighbourhood of S .The discussion in Section 3 shows that the electromagnetic field(21) F = h ( α , β ) ∇α × ∇β has a set of closed electric field lines in the shape of the mirror image of S i ∈ I L i and a setof closed magnetic field lines in the shape of the mirror image of S i / ∈ I L i at time t = F contains these links for all time, whichconcludes the proof of Theorem 1.1, since every link has a mirror image. (cid:3)
5. D
ISCUSSION
We showed that every link type arises as a set of stable electric and magnetic field linesin a null electromagnetic field. Since these fields are obtained via Bateman’s construction, they share some properties with the torus link fields in [19]. They are for example shear-free and have finite energy.However, since the proof Theorem 1.1 only asserts the existence of such fields, via theexistence of a holomorphic function h , other desirable properties of the fields in [19] aremore difficult to investigate. The electric and magnetic field lines in [19] lie on the levelsets of Im ( α p β q ) and Re ( α p β q ) . At this moment, it is not clear (and doubtful) if thefields in Theorem 1.1 have a similar integrability property. It is, however, very interestingthat the relevant function z p z q , whose real/imaginary part is constant on integral curves ofthe (pushforward of the) magnetic/electric field, is (up to an added constant) exactly thecomplex plane curve whose totally tangential intersection with S gives the ( p , − q ) -toruslink. In light of this observation, we might conjecture about the fields in Theorem 1.1,which contain L , that if the electric/magnetic field lines really lie on the level sets of a pairof real functions, then the real and imaginary parts of F would be natural candidates forsuch functions, where F = S totally tangentially in the mirror image of L . Sofar z p z q − = ( p , − q ) -torus link) that the author is aware of, even though it is known to exist for any link. It is thislack of explicit examples and concrete constructions that makes it difficult to investigatethis conjecture and other properties of the fields from Theorem 1.1.Kedia et al. also obtained concrete expressions for the helicity of their fields [19].Again, the lack of concrete examples makes it difficult to obtain analogous results.Since the fields in Theorem 1.1 are obtained via Bateman’s construction, all their Poynt-ing fields at t = α and β to obtain knottedfields, whose underlying Poynting fields give more general Seifert fibrations.R EFERENCES[1] C. C. Adams.
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