However, I can’t think how this could lead to causality-violating effects like time travel or anything else — which would seem to be necessary if you were trying to show that deja-vu was anything more than a purely cognitive phenomenon. So no, there’s no-one working on these things!

Also, your caps-lock key seems a little broken. Thanks for the comment!

Building on the interest in my post on knots and different dimensions, I thought I’d say a few words on some in [...]

I must actually confess, I’ve not done any research on the topic, but this is what I think: one can show (unless I’m missing something subtle) that a plane is indeed the solution to my problem. One can visualise it as a string in 3 dimensions, extended infinitely in the “time” direction (but it’s still a spacial direction we’re visualising as time, so that permutations of the knot are made simultaneously at all points along the “time” spacial direction). Then my untying trick won’t work.

And that is also, I believe, a proof of the Cook conjecture you offer above. Treat the first three dimensions as though they were a normal string being tied in 3 spacial directions. Then treat all subsequent dimensions as dimensions of infinite extension of the “string”, so that they can’t be used to untie anything — the knot exists at all points in those directions.

From your explanation of how to untie a four dimensional knot, an object that couldn’t be untied in that way would be a three dimensional string that is there for all time. This forms a two dimensional plane in four dimensions. Is this a general pattern - in N dimensions you can knot only N-2 dimensional objects? Seems like it should be true, but I wouldn’t know how to prove it.

peace.
holly