No, first of all, because such knots describe the topology of propagating electromagnetic  fields, which is what the solutions to Maxwell's equations are. These are electromagnetic waves, whose field lines, while closed (standard property of electromagnetic waves) are, also, knotted. That's all. Second, we do know that dark matter isn't  ordinary  matter-that label includes the mediators of the known forces, the photon being the mediator of the electromagnetic interactions. 

I am sure I am naive, but as far as I know, all that is known about dark matter is that a lot of gravitational mass exists which is non-interacting, or very weakly interacting..  The E-M knots are propagating, but the Pointing vector describes closed orbits.  i.e., I think that there is no linear propagation.  The surprising thing is that Maxwell's equations permit such solutions.  I believe that your second point is simply incorrect.  We know NOTHING about dark matter, neither what it is nor what it is not, except that it exerts gravitational force.

EM knots are electromagnetic radiation-that's what's relevant. So, if they were a component of dark matter, they would have given a signal, that hasn't been observed. While we do not know the nature of the particles, that compose dark matter, we do know that they are not known particles, because we can compute the consequences of known particles, that have, in addition, the requisite gravitational interactions, and the result is that the consequences are incompatible with the observations. Electromagnetic waves, that would give rise to such gravitational interactions, would have been detected readily-it is known how to detect electromagnetic signals. This, also, places constraints on the electromagnetic interactions of dark matter particles. 

If dark matter has *only* gravitational interactions with known matter, then detecting its particle content will be very hard. That's why the dark matter detectors assume that dark matter particles have very weak interactions, but stronger than gravitational, with known particles and try to place limits that way, since, then, they will have a better chance of detecting the decay products, which are normal particles, of course. 

A knot of EM energy has no linear motion.  So how does the knot transmit a signal to anything outside the Knot?  As far as I can "reason" the knot does not communicate its exstence to anything outside the knot.  Please explain how the knot, which is limited in spatial extent, can be observed.

The electric and magnetic fields, solutions of Maxwell's equations, describe closed field lines, of course, in the absence of sources.    What wasn't noticed, apparently, till now, was that these lines-loops in three-dimensional space-could be knotted. That's all. So one has a traveling wave, with knotted field lines.  The lines can't (self) intersect, but nothing prevents them from being knotted. Cf., for example: ``Linked and knotted beams of light, conservation of helicity and the flow of null electromagnetic fields'', http://arxiv.org/abs/1110.5408

The direction of propagation doesn't have to be along a  loop (knotted or not) itself, however.

In terms of mechanics, does the E/M knot transport energy in any  LINEAR direction, or can the energy of the knot stay localized?  If the former, is the case, it is akin to light, and can be observed.  If the latter is true (and this was my reading of the Poynting vector) the energy can stay localized, and such knots cannot be observed.  If the energy of a knot can stay localized, and the time average of the Poynting vector is zero, it is a candidate for dark matter. 

Of course it can, like any electromagnetic wave. Once more: the knot is a property of any loop-and the field lines of the electric and magnetic fields, in the absence of sources, that can describe propagating waves, are loops. To date  unknotted loops were considered, but that's not necessary; knotted loops can also describe the electric and magnetic field lines of propagating waves. Such loops are, necessarily, non-planar (though non-planar loops aren't, necessarily, knotted, of course) but that's not a physical limitation-there isn't any constraint from Maxwell's equations, that implies that field lines must be *planar* loops. Mathematically one just uses Frénet frames to describe the loops. . The Poynting vector is defined as usual.So, of course, it describes energy transfer. The knots don't stay put, they move along as an electromagnetic wave (cf. irvine's paper I cited, for an example).

In any event, other charged particles, of ordinary matter, moving along such electric field lines, or spiraling about the magnetic field lines, would radiate in turn , just as would be the case for any electromagnetic field, and this radiation would have been observed. So knotted electromagnetic fields can't describe dark matter. Not to mention the fact that electric and magnetic fields have charged particles as sources, that would be expected to give rise to other electromagnetic field configurations, besides knots, and it's known that the electromagnetic response of dark matter is very small. 

I request your patience, but my key question has not yet been addressed.  I understand electromagnetism quite well.  My questions are specific, and require specific responses, not answers of a general or complete nature.  I will capitalize the CORE of each question.  Can a knot be LOCALIZED in space?  I am not asking what the typical knot does, but can a SUB-GROUP of knots exist without average linear motion?  Of course such a class of knots, if they exist, are not plane waves.  Why bring that self evident point up?  The question is not about knots in general, but specific to the existence of SOME knots without average linear displacement.   Can a SUBCLASS of knots exist which have no AVERAGE LINEAR motion? 

Regarding the second paragraph of your most recent response:  Charged particles do not move along electric field lines.  They are deflected from their initial path by the Lorentz force.  In particular, ions  of mass equal to the proton, or approximately a multiple thereof, having even low momentum are very difficult to deflect, and when deflected have negligible bremstrahlung radiation.   Electrons are easier to deflect, and much more likely to emit bremstrahlung, but an argument based upon detectable electron bremstrahlung depends critically on the answer to the next order of questions: If quasi-stationary E/M knots CAN exist, what limitations exist on the field  magnitudes?

Sorry if I wasn't precise enough-but the basic point is that ordinary charges, subject to such fields will radiate.

To get back to your question: I think that the question about localization of a knot distracts from the key issue. Take as an example the magnetic field produced by a current carrying wire. The field lines are circles, loops, that are unknotted. If the current varies in time, there will be travelling waves, with closed field lines; the amplitude of the field  will be controlled by the initial value data. Now it's possible to imagine-and Irvine and Kedia give examples-of field line configurations that are knotted loops. These are admissible initial value data (satisfying the constraint equations) and define here knots, whose size is controlled by the value of the initial field configurations. The knotted property depends on the sources; the size of the knot isn't fixed, just like the radius of the circles for the wire isn't fixed. That's because Maxwell's equations have conformal invariance-so no scale is preferred, unless it's defined by the source.

Nothing in this analysis depends on the knottedness-that's the point, what's crucial is that the field lines are loops and loops in three-dimensional space can be knotted-but this word is about mathematics, not physics. The physics is controlled by the wave equations, subject to, simply, unusual initial conditions. 

You stlll avoid my direct question.  In anE/M knot in space, where there is no wire, and there are no oscillating electrons, and there is no source of traveling waves - in that charge-less space,  can a knot exist which is NOT part of a traveling wave?  Traveling waves self propagate.  If a knot can exist without an associated traveling wave, the Poynting vector average would be zero, and it would have zero liniar displacement.  So, my stripped down question is - can an E/M knot exist in space, not associated with a traveling wave?.

Do Maxwells equations permit solutions such as an E/M knot in EMPTY space, which is not part of a traveling wave?

The knot, as such, is irrelevant-the same question holds for any electromagnetic field configuration and the answer is, of course, the same. There isn't any such thing as an associated wave, of course-the electromagnetic field configuration, knotted, or not, can, indeed, be described by a traveling wave, if the appropriate initial and boundary conditions are satisfied. The knot describes the topology of the lines, that's all.  A knot is the result of a particular source, just like a circle is the result of a particular source and the fields on them  both vanish, if the source vanishes and the boundary conditions are compatible with it-Maxwell's equations are linear equations. 

I appreciate your effort, and I will quit after this note, since I am unable toclarify my question.  But, here is my final try.  A plane wave in spare is a solution to Maxwell's equations, without reference to source, origin or destination.  Can the same be said about the possibility of a stand alone knot?

It's not true that a plane wave in space is without reference to a source,  origin and destination, i.e. initial and boundary conditions; the solution to Maxwell's equations, with vanishing sources and fields vanishing at infinity, for instance, is that where all fields vanish everywhere. A plane wave is the solution of Maxwell's equations, subject to certain initial and boundary conditions. A knot is in all aspects similar, just these conditions are different.  

Stam, you prevent me from asking a question concerning a possible physical state because you are insisting on general solutions.  A plane wave in space IS a solution to Maxwells equations, as published.  If you ask "how did it get there", you need a source.  I am trying to pose a question which strips out all conditions which can be stripped out.  If an oscillating electron group manages to emit a plane wave. and then is annihilated by a similar group of positrons, the source is gone, but the radiation persists.  I am able to ask questions (i.e., do experiments) on that plane wave without concern that the latter part of it will be distorted by the annihilation effects.  It is a physical state subject to measurement.    

The same fate will befall the knot: if the source that emits it is ``annihilated'', as you describe, the knot will nonetheless travel along and will be detected. It's in no way different from any other waveform. (Technically, of course, the photons from the annihilation will, also, arrive, but I agree with you.) There's nothing special about a knot. 

Are you familiar with the work by Albert Einstein, circa 1955, on (if I remember correctly) self binding E/M masses named "geons"?  If my memory serves (questionable, at my age) these were E/M clusters having energy, and hence mass. which satisfied Maxwells equations and the equations of general relativity.  My memory is that these clusters could behave as stationary masses, although they were composed of only E/M fields.  Modulo my functioning memory, these properties appear to violate your arguments concerning stationary knots.

No-but that's a totally different system, because it involves the coupled equations of general relativity and electromagnetism. Of course their solutions   can have stationary masses-namely, those of the corresponding Reissner-Nordstrom black holes. But they don't contradict any arguments about pure electromagnetism, because Maxwell's equations are linear and are invariant under conformal transformations (up to the source terms). In four spacetime dimensions, incidentally, there are uniqueness theorems for the black hole solutions, that fail in higher dimensions. 

Stam, The world involves coupled equations.  It's the system we live in.  That's what physics is all about.

Of course, but there are consistent approximation schemes-that can be quantified. That's, also, part of physics-to recognize which effects are dominant when. But it's interesting what fascination a word like ``knot'' can bring forth-whereas it's nothing different than a particular electromagnetic field configuration, for which everything known applies-cf. Irvine's paper.

Stam,

You mistake open-mindedness - a prerequisite for extension of knowledge boundaries - for fascination.  In our rather long exchange of questions and answers, I did not get an answer to my initial question.  That is one result of dedication to current approximations.

Once more: the answer to the original question is No, the knots in the electromagnetic field can't account for dark matter. And this statement holds, independently of the possibility that electromagnetic field configurations may be knotted. Electromagnetic fields, furthermore, possess energy but not mass-they have gravitational effects because the metric couples to the energy-momentum tensor in general relativity. However the mass, as defined in general relativity, of photons, the excitations of the electromagnetic field, is zero.  Finally, static configurations of finite energy don't exist in electromagnetism-so it's not true that knots wouldn't emit light, i.e. electromagnetic radiation, either. The only classical,  static, configurations would be Reissner-Nordstrom black holes (which would emit Hawking radiation, as long as their mass exceeded their charge). However black holes can be detected by studying how matter moves in and radiation is emitted from their vicinity.