It seems that most of the time these days, I being asked to think about things in a different number of spacial dimensions than 3 (up-down, left-right, forwards-backwards). Now, some properties follow through to additional dimensions quite easily, but I was thinking recently about one that doesn’t: knots — it’s a great example of the tricks that dimensions can play on one.
What’s a knot? Take a piece of string (an object extended in one dimension). Form a loop and feed it through the loop. The knot so formed cannot, by continuous deformation of the string, be “undone” without moving an end of the string back through the loop.
But knots only exist in three spacial dimensions.
Try two dimensions (which can be simply drawn on a flat piece of paper). Forming the loop leaves a diagram that is indistinguishable from a straight piece of string plus a seperate loop, which is free to float away. So you no longer have one piece of string.
Now, try four dimensions. The best way to try to visualise this is to treat time as a fourth spacial dimension; that is, any particular piece of string only exists at one instant in time, and it is possible to move the different parts of the string forwards and backwards in time, so long as the string remains continuous.
Ok, form the entire knot at one instant in time. Now move the side of the string that went through the loop, forward a litle in time, so the rest of the knot is not there. Move it spacially a bit such that when you move it back in time to the rest of the knot, the string no longer goes through the loop. You’ve just continuously transformed the knot into a non-knot (as well as given yourself a headache, if my experience is any indication).
So, knots only exist in 3 spacial dimensions.
Exercise for the reader who hasn’t had enough yet: Is there a structure that in 4 spacial dimensions can be tied into the equivalent of a knot?
7 thoughts on “Knots and different dimensions”
the spacial dimension of knots and the unrest in togo…paul, you sure write about interesting stuff. i don’t know anyone else on the planet who devotes their blog postings such subjects. it is fabulous. what can i say? i’m happy to have cool neighbors. if i need help tying knots or tips on how to overthrow a government, i know where to go!!!
Maybe if we embedded Togo in four dimensions, stuck a few crosscaps on it in choice places and pulled it back into three space, we could invert the members of the junta thereby making it completely impossible for them to function in a right-handed coordinate frame. Retaking the capital would be easy after that.
Very nice problem to think about. I don’t know about the string, but my mind certainly feels pretzel-shaped now. Interesting thought – all the four dimensions should be equivalent, so you should be able to visualise any one of them as time and the other three as as spatial. If you take a three dimensional knot then, add a dimension, and then choose a different three dimensions as spatial, then I guess in general it wouldn’t even look like a knot.
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.
Thanks, I’m pleased you enjoyed it!
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.
have you ever been out on a hot summer day and found yourself suddenly in the midst of a dust devil?SOME ARE LARGE ENOUGH TO PACK A SMALL PUNCH. WELL,WHAT IF SOMEHOW THERE ARE SPACIAL’ DEMENSIONAL ,CHANGES AOURND US ,AND JUST LIKE PHYSICAL ATMOSPHERICAL CHANGES CAN CAUSE CHANGES THAT WE CAN MEASURE AND OBSERVE,THERE ARE CHANGES IN TIME AND SPACE AND DEMENSIONS THAT WE HAVE NOT YET LEARNED TO INTERPRET OR OBSERVE ,BUT HAVE SYMPTONS LIKE DEJAVU,OR MEETING SOMEONE THAT YOU INSTANTLY KNOW ,BUT KNOW THAT YOU COULD NOT HAVE EVER MET THAT PERSON IN THE COURSE OF THE LIFE THAT YOU RECALL?IS THERE ANYONE DOING STUDIES ON THESE THEORIES?
It’s certainly possible that we could observe the effects of additional dimensions, by for example observing the formation of black holes at lower energies than one would naively expect in particle accelerators.
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!
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