Understanding Knots in Higher Dimensions

[oldman]

While re-reading James Jean's The Mysterious Universe I came across his statement that "the capacity for tying knots is limited to space of three dimensions", in the context of sailor's knots. I'm no mathematician, but understand that mathematical folk regard a knot more generally as a topological feature of an n-dimensional closed loop "embedded in" an m (>n+2)-dimensional space.

May I ask why the 2?

Seems reasonable to me for ropes in the space we live in. Ropes are like 1-dimensional lines that need one extra dimension to be loops in and another extra dimension to be crossed into make knots. Makes a total of 1+2 = 3 needed dimensions. Does having more dimensions than 3 for ropes simply mean that the crossings can then be removed by using yet other dimension(s) to undo crossings without cutting the loop or passing the rope through itself? Or is this too simple minded?

Somebody must somewhere have suggested that knotted loops, perhaps unrecognised as such, might be enduring features in the three dimensions of the spacetime we inhabit. Probably in Maxwell's time.

 

[slider142]

All closed curves in dimensions greater than 3 are homotopic to S1, the circle. As an example, note that you need only add color to add a 4th dimension to a knot in 3-dimensional space. You can then simply pass two pieces of different color "through" each other, as they occupy a different fourth coordinate.
Knot theory in higher dimensions necessarily studies higher-dimensional analogues to closed loops. In dimension n, one studies (n-2)-dimensional knots.

 

[oldman]

All closed curves in dimensions greater than 3 are homotopic to S1, the circle. As an example, note that you need only add color to add a 4th dimension to a knot in 3-dimensional space. You can then simply pass two pieces of different color "through" each other, as they occupy a different fourth coordinate.
Knot theory in higher dimensions necessarily studies higher-dimensional analogues to closed loops. In dimension n, one studies (n-2)-dimensional knots.

Thanks for the reply. I liked your colour example, but sadly I don't even know what homotopic means. Also, it's nice to know that "In dimension n, one studies (n-2)-dimensional knots" --- but, still, why the 2? You're talking to a mathematical simpleton here -- one who doesn't even appreciate the current physics fashion of talking in acronyms like LBNL, ADEPT and BAO. Maths jargon is also difficult for me!

 

[Tinyboss]

Rather than homotopy, knots are usually classified up to "ambient isotopy", which roughly means that if you can smoothly deform the space the knot occupies, and in the process transform that knot to some other knot, then the two knots are equivalent.

Intuitively, it's just what you would do if you had an actual loop of string and were pushing and pulling it, trying to get it into a particular shape.

 

[oldman]

... knots are usually classified up to "ambient isotopy"

Intuitively, it's just what you would do if you had an actual loop of string and were pushing and pulling it, trying to get it into a particular shape.

Thanks for clarifying what is meant by homotopy. I understand what you say, but it's always needing two extra dimensions for making knots possible that puzzles me, no matter how many (10, 11, 23?) space dimensions there are. And the fact that we seem to live in the apparently minimalist situation of just three space dimensions.

 

[slider142]

The color example is the base problem. It is simply that if you have 3 extra dimensions of freedom, you have too much freedom to create a knot. The problem is straightforward: take any knot in a 3-dimensional space, and now add one extra dimension as color. The knot is all one color to start with, as the fourth coordinate is some constant. Now we can simply smoothly change the color in some small region of the knot (just like lifting a string out of a plane, we change the fourth coordinate by lifting the color) and pass unlike colors through each other to undo the knot (since they do not have the same fourth coordinate, they are not actually passing through each other in 4 dimensional space). We know we can do this because every closed curve that does not intersect itself is homeomorphic to a circle (a basic theorem of topology).
The general argument for closed curves in higher dimensions can be reduced to the previous example.
1 additional dimension does not allow paths to cross (ie., think of trying to create knots out of 1-dimensional circles in a 2-dimensional plane). 2 additional dimensions is just right: it allows us to pass paths around each other, but does not allow so much freedom that we can unknot any such path.

 

[lavinia]

While re-reading James Jean's The Mysterious Universe I came across his statement that "the capacity for tying knots is limited to space of three dimensions", in the context of sailor's knots. I'm no mathematician, but understand that mathematical folk regard a knot more generally as a topological feature of an n-dimensional closed loop "embedded in" an m (>n+2)-dimensional space.

May I ask why the 2?

Seems reasonable to me for ropes in the space we live in. Ropes are like 1-dimensional lines that need one extra dimension to be loops in and another extra dimension to be crossed into make knots. Makes a total of 1+2 = 3 needed dimensions. Does having more dimensions than 3 for ropes simply mean that the crossings can then be removed by using yet other dimension(s) to undo crossings without cutting the loop or passing the rope through itself? Or is this too simple minded?

Somebody must somewhere have suggested that knotted loops, perhaps unrecognised as such, might be enduring features in the three dimensions of the spacetime we inhabit. Probably in Maxwell's time.

In higher dimensions - 4 or more - two loops can not be linked. They can always be separated. Also any knot can be unknotted.

 

[oldman]

...
The general argument for closed curves in higher dimensions can be reduced to the previous example.
1 additional dimension does not allow paths to cross (ie., think of trying to create knots out of 1-dimensional circles in a 2-dimensional plane). 2 additional dimensions is just right: it allows us to pass paths around each other, but does not allow so much freedom that we can unknot any such path.

Thanks again, Slider, for this extended reply. Colour is a nice imagineable extra dimension. The reason for needing two extra dimensions, but no more, to create a knot is simple, as I guessed in my O.P., for a knot in ordinary string , itself something essentially one dimensional albeit flexible in two dimensions.

You refer to "closed curves in higher dimensions (that) can be reduced to the previous example (of string)." Are these "curves" you refer to one-dimensional, or in higher m-dimensional spaces are they {n (>1) = m-2} dimensional? i.e. themselves multi-dimensional, like say bulked-up rope or something else unimaginable for us 3-D creatures.

I thought this was the case and the 2 puzzled me for the general case, but perhaps I'm wrong. If not, I'll just have to take somewhat ex-cathedra pronouncements, like that of ]Livinia, on trust.

 

[slider142]

You refer to "closed curves in higher dimensions (that) can be reduced to the previous example (of string)." Are these "curves" you refer to one-dimensional, or in higher m-dimensional spaces are they {n (>1) = m-2} dimensional? i.e. themselves multi-dimensional, like say bulked-up rope or something else unimaginable for us 3-D creatures.

I thought this was the case and the 2 puzzled me for the general case, but perhaps I'm wrong. If not, I'll just have to take somewhat ex-cathedra pronouncements, like that of ]Livinia, on trust.

The generalized "curves" refer to any topological sphere embedded in a higher dimensional space. By the previous theorem, we see that knots in n-dimensional space are necessarily (n - 2)-dimensional.

 

[lavinia]

The color example is the base problem. It is simply that if you have 3 extra dimensions of freedom, you have too much freedom to create a knot. The problem is straightforward: take any knot in a 3-dimensional space, and now add one extra dimension as color. The knot is all one color to start with, as the fourth coordinate is some constant. Now we can simply smoothly change the color in some small region of the knot (just like lifting a string out of a plane, we change the fourth coordinate by lifting the color) and pass unlike colors through each other to undo the knot (since they do not have the same fourth coordinate, they are not actually passing through each other in 4 dimensional space). We know we can do this because every closed curve that does not intersect itself is homeomorphic to a circle (a basic theorem of topology).
The general argument for closed curves in higher dimensions can be reduced to the previous example.
1 additional dimension does not allow paths to cross (ie., think of trying to create knots out of 1-dimensional circles in a 2-dimensional plane). 2 additional dimensions is just right: it allows us to pass paths around each other, but does not allow so much freedom that we can unknot any such path.

Going the other way seems hard to me. can you explain it? I can't easily see why a knot in R^4 can be untied. Your color argument would work if you could first show that the knot can be deformed without self intersection into R^3. Is that possible?

 

[oldman]

The generalized "curves" refer to any topological sphere embedded in a higher dimensional space. By the previous theorem, we see that knots in n-dimensional space are necessarily (n - 2)-dimensional.

Please let me check that I finally do follow your clarifications, Slider.

Knots in ordinary string loops are called by topologists "knots in closed curves that are 1-dimensional topological spheres (a.k.a. "circles") embedded in 3-dimensional space" . Similarly, in higher dimensions, topologists would talk of "knots in closed curves as with (for example) 5-dimensional topological spheres embedded in 7-dimensional space" . The "curves" are themselves multi-dimensional.

And, in for example 7-dimensional space, the only possible knots are 5-dimensional entities. Such knots could not exist in any space other than one with 7 dimensions. Have I got this right?

Knots in a closed curve of any dimensionality always involve two extra dimensions. Is thinking of one to "curve" in, the other to allow the "curve" to "cross" itself too simplistic? Topologists have established, it seems, that even a single extra dimensions is one too many for knots. It allows all crossings (each involving a minimum of three dimensions?) to be eliminated, somehow, so rendering knots impossible.

Finally to persist in horse-fly mode (my apologies for this), what "previous theorem" were you referring to in the above quote?