Answer to the Question 04/98

KNOTTED FIELDS

The question was:

Field lines of magnetic fields are always closed loops. Does there exist a configuration of currents which creates a KNOTTED line of a magnetic field.



(4/98) It has been been observed by several people, that the question, in the form in which it was presented, has a trivial solution:

Consider a knotted line in three-dimensional space (trefoil knot, granny knot, or knot of any topology you like). Take a long narrow solenoid and fold it into the shape of the knot. Connect the ends of the solenoid (and, of course, insert a power source somewhere along the line). Inside the solenoid there will be magnetic field lines parallel to its axis, and therefore those lines will form a knot.

So we see that the solution is simple. HOWEVER, we notice the current which formed the knotted magnetic field line is by itself knotted. Therefore, we would like to ask the follow-up question:

Is it possible to create a knotted field line by a linear current loop which is NOT KNOTTED?

(6/98) Christian von Ferber (e-mail ferber@orion.tau.ac.il), a post-doc at Tel Aviv U., correctly solved the follow-up problem. He had shown that one by slight modification of the construction which was explained above it is possible to create knotted field by unknotted currents. Here is what he wrote:

Assume we have formed a trefoil knot with our solenoid with the knot topology given by the sketch below. The magnetic field lines are then obviously knotted.

We will now unknot the wire without changing the topology of the magnetic field line along the center of the solenoid. For this note that we only have to change one 'bridge' in the sketch above to a 'tunnel'.

Imagine a blow-up of one bridge. We pull apart a little bit the two crossing parts of the solenoid to illustrate the bridge formed by the thin wire:

We may then unknot the wire while not moving other parts of the solenoid:

Finally we push together the solenoid again keeping the changed topology of the wire. Thus the centerline of the solenoid is unchanged in the form of a trefoil knot, while the wire itself is of zero-knot topology.

(7/98) Y. Kantor: After publication of the solution we received an e-mail from Michael Trott (e-mail mtrott@wolfram.com) from Wolfram Research, Inc. claiming that "It is wrong belief that magnetic field lines are closed! Typically they are not closed." The discussion between Trott, and v. Ferber on the subject followed, and is presented here. Since the argument was partially numerical, I do not feel that a firm proof of either position has been presented. I therefore present the entire argument to you. Further comments are welcome!

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