Suppose we have a rigid token-ring network. An observer at any node can seemingly determine the angular momentum of the network by measuring the time it takes for a packet to travel around the ring in each of the two directions. Is it possible by any means for an observer to determine whether the network is knotted?

  • $\begingroup$ Its a loop right? So how can a packet take different time in either direction? $\endgroup$ – TROLLHUNTER Mar 9 '11 at 3:38
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    $\begingroup$ Because of special relativity. See en.wikipedia.org/wiki/Ehrenfest_paradox $\endgroup$ – Dan Brumleve Mar 9 '11 at 3:51
  • $\begingroup$ Ok, so what information do we have, only the two times? Can we measure anything else? $\endgroup$ – TROLLHUNTER Mar 9 '11 at 4:08
  • $\begingroup$ Sure, if the packet is an electron instead of a photon then the loop has a current and we can measure a magnetic field somewhere. I guess this would tell us something about how coiled up it is but is there any information about the knot? $\endgroup$ – Dan Brumleve Mar 9 '11 at 4:12
  • $\begingroup$ I am intrigued by this question. A token ring is a telecoms entity, I assume. But what does "knotted" mean? $\endgroup$ – Roy Simpson Mar 10 '11 at 22:53

Actually, an observer can't measure even the angular momentum. For example, suppose the token ring is in the form of a figure 8. Then the times around the two directions are identical even if the token ring is rotating.

  • $\begingroup$ Indeed. Maybe it would make sense if each node had information about the angle formed by its adjacent neighbors? $\endgroup$ – Dan Brumleve Mar 13 '11 at 22:58

Measure electric field. This gives you position of electron. And complete topology of wire.

  • $\begingroup$ Where and when to measure? We already know the topology of the network is a loop but we need to know the topology of the volume outside the network. $\endgroup$ – Dan Brumleve Mar 9 '11 at 5:46
  • $\begingroup$ @Dan Im not totally sure I get the question, but if you measure the electric field at any point, you can determine where the electron is at any time, and find the path it travels. So if it travels in a knot, the network is knotted. $\endgroup$ – TROLLHUNTER Mar 9 '11 at 5:49
  • $\begingroup$ I know the question is obtuse and I appreciate your effort and patience. I am thinking of the electron packet as a DC current through e.g. a copper wire. Measuring its resistance gives the same information as the case of a photon in free space. Measuring the magnetic field at some points a distance away from a node gives additional information (the winding number of the path?). But the electric field is everywhere absent because the current-carrying wire is neutral. $\endgroup$ – Dan Brumleve Mar 9 '11 at 6:02
  • $\begingroup$ @Dan Well you can measure the gravitational field instead of electric field. Then one can in theory find the exact path of the electron. $\endgroup$ – TROLLHUNTER Mar 10 '11 at 23:10

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