August 30, 2011 10:58 PM Subscribe

Mathematics/Topology: how to untie a trefoil knot in four dimensions?

To my general understanding all 3d knots are not really knots four dimensions; practically I thought this simply involved looking at the process of tying the knot over time, and observing that the process is reversible -- that there is thus no point in 4d at which the simple string "unknot" is fundamentally changed to a knot.

However when I looked at the trefoil knot this understanding fell to pieces. Obviously there is no process to build a trefoil that starts from an untied string without changing its topology fundamentally (by closing the ends together).

Hence my question: probably the trefoil knot is still no knot in 4d, but how so? Can you point me to any resource which helps visualizing the topology of a trefoil in 4d?
posted by knz to Science & Nature (10 answers total) 4 users marked this as a favorite

To my general understanding all 3d knots are not really knots four dimensions; practically I thought this simply involved looking at the process of tying the knot over time, and observing that the process is reversible -- that there is thus no point in 4d at which the simple string "unknot" is fundamentally changed to a knot.

However when I looked at the trefoil knot this understanding fell to pieces. Obviously there is no process to build a trefoil that starts from an untied string without changing its topology fundamentally (by closing the ends together).

Hence my question: probably the trefoil knot is still no knot in 4d, but how so? Can you point me to any resource which helps visualizing the topology of a trefoil in 4d?

BP linked to some good pictures that help explain how one would do it, but looking at your question there seems to be a more fundamental confusion.
^{3}.

posted by JiBB at 11:16 PM on August 30, 2011

But the unknot isn't a piece of string with two ends, it's a closed loop, an embedding of a circle into RObviously there is no process to build a trefoil that starts from an untied string without changing its topology fundamentally (by closing the ends together).

posted by JiBB at 11:16 PM on August 30, 2011

It might help to use a similar visualization to understand an impossible-in-2D task in 3D.

Imagine that in the 2D plane you have a square inside of a circle, and you're trying to move the square outside of the circle without the two crossing. Using the same visualization as in BP's link of color representing movement into an additional, orthogonal, dimension, we can visualize things moved up as being more green. Move the square up (color it green). Now you can move it outside of the circle without them crossing. Then, move it back into the original plane (color it white again).

posted by JiBB at 11:24 PM on August 30, 2011

Imagine that in the 2D plane you have a square inside of a circle, and you're trying to move the square outside of the circle without the two crossing. Using the same visualization as in BP's link of color representing movement into an additional, orthogonal, dimension, we can visualize things moved up as being more green. Move the square up (color it green). Now you can move it outside of the circle without them crossing. Then, move it back into the original plane (color it white again).

posted by JiBB at 11:24 PM on August 30, 2011

We can't easily visualize objects in four dimensions, because we exist in a three-dimensional reality (at least within the scale that humans exist in) and we evolved a visual or cognitive language for describing and understanding 3D objects and how to manipulate them.

The best we can manage is a

posted by Blazecock Pileon at 11:31 PM on August 30, 2011

First of all, forget about the idea that time is the fourth dimension. Time is *a* fourth dimension, sure, and if we were talking about physics we might say "4D" and mean four-dimensional space-time, but if we're talking about topology, "4D" means four spatial dimensions. And that's an area where we can get the best understanding by using analogies from the relationship of 2D to 3D.

Consider a closed 2D figure lying in a plane. Within that plane, that figure has a definite inside and a definite outside, and if it's a barrier to motion, then an object on the inside would be confined there, like a ball inside a billiards rack. But if that object is allowed to move through three dimensions, it can move from the inside to the outside by leaving the plane entirely, right? Just pick the ball off the table and place it down again outside the rack. While it's completely outside the plane, it can't collide with a figure completely inside it.

Similarly, a closed surface in 3D space (say, a sphere) has an inside and an outside, and an object on the inside can't move to the outside without breaching the sphere. But if we had access to a fourth dimension, we could move the object perpendicular to the 3D space and then put it back in outside of the sphere. While it's outside of our 3D space, it's effectively intangible to anything inside it.

Now, having understood that, the easiest way to understand untying a knot through the fourth dimension is to imagine leaving most of the knot in its 3D space, but stretching a piece of it into the fourth dimension, similar to how you'd pluck up one part of a string lying on a table. While that bit of the knot is outside of the 3D space, it is incapable of touching the bits that are inside it. So we can just shift it over to the other side of some strand of the knot and then plunk it back into the 3D space, just like we shifted the billiard ball to the other side of the rack.

Now, if you really want to break your brain, consider this: Even though 4D space doesn't admit knotted curves, it does create the possibility of knotted*surfaces*. Visualizing those is left as an exercise for the reader.

posted by baf at 1:05 AM on August 31, 2011 [7 favorites]

Consider a closed 2D figure lying in a plane. Within that plane, that figure has a definite inside and a definite outside, and if it's a barrier to motion, then an object on the inside would be confined there, like a ball inside a billiards rack. But if that object is allowed to move through three dimensions, it can move from the inside to the outside by leaving the plane entirely, right? Just pick the ball off the table and place it down again outside the rack. While it's completely outside the plane, it can't collide with a figure completely inside it.

Similarly, a closed surface in 3D space (say, a sphere) has an inside and an outside, and an object on the inside can't move to the outside without breaching the sphere. But if we had access to a fourth dimension, we could move the object perpendicular to the 3D space and then put it back in outside of the sphere. While it's outside of our 3D space, it's effectively intangible to anything inside it.

Now, having understood that, the easiest way to understand untying a knot through the fourth dimension is to imagine leaving most of the knot in its 3D space, but stretching a piece of it into the fourth dimension, similar to how you'd pluck up one part of a string lying on a table. While that bit of the knot is outside of the 3D space, it is incapable of touching the bits that are inside it. So we can just shift it over to the other side of some strand of the knot and then plunk it back into the 3D space, just like we shifted the billiard ball to the other side of the rack.

Now, if you really want to break your brain, consider this: Even though 4D space doesn't admit knotted curves, it does create the possibility of knotted

posted by baf at 1:05 AM on August 31, 2011 [7 favorites]

I think my brain just exploded. But that was one of the coolest illustrations I've ever seen.

posted by kathrynm at 4:51 AM on August 31, 2011

posted by kathrynm at 4:51 AM on August 31, 2011

I like the idea of knotted surfaces; what would a 3d projection of such a beast look like? Did anyone try this already?

posted by knz at 12:25 PM on August 31, 2011

posted by knz at 12:25 PM on August 31, 2011

Clifford Stoll sells 3D projections of Klein bottles.

posted by Blazecock Pileon at 12:00 AM on September 1, 2011

This thread is closed to new comments.

posted by Blazecock Pileon at 11:03 PM on August 30, 2011 [1 favorite]