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Can planes be tied in knots in higher dimensions the way lines can in 3D? (askamathematician.com)
92 points by joeyespo on Feb 28, 2016 | hide | past | web | favorite | 51 comments

As a matter of fact, yes, planes can be tied in knots.

Here's a page from "Knotted Surfaces and Their Diagrams" by J. Scott Carter: http://i.imgur.com/2rIKJ8E.png

Need someone's insight: Isn't klein bottle a 3D knot?

Knots are typically seen as embeddings (non self-intersecting image) of S^n, a closed orientable surface. For example, S^1 (the circle) in R^3 (3-dimensional space) is what we usually think of as a knot. The simplest knot, trefoil knot, is just a circle embedded into the ambient space in a way that cannot but "untangled" without self-intersection.

The Klein bottle, being non-orientable, is not usually considered a knot.

But the important thing is that a knot is all about the ambient space. If you are allowed 4-dimensional motions, then you can always untangle any embedding of S^1.

No, a Klein bottle is a 4d moebius strip.

A Klein bottle is a 2-dimensional manifold. It can't be embedded in R3, but it can be embedded in R4. This is not what people normally mean by "4d", though. It is like a Moebius strip in that they are both 2-dimensional non-orientable manifolds, though of course a Moebius strip can be embedded in R3. The Moebius strip has a boundary, which sometimes excludes it from a strict definition of "manifold".

A Möbius strip has a boundary. What is the boundary of a Klein bottle?

You can construct a Klein bottle using a rectangle with two edges with the same orientation and two edges with opposite orientation (K^2 is a good notation for this). Naturally identifying the same orientation edges gets you a cylinder. When you identify the remaining two edges in R^3 you have the classic Klein bottle we're familiar with. But there is no embedding of K^2 into R^3, but there is for R^4. This is intuitive because the map from K^2 to R^3 is not one-to-one because two circles of K^2 have the same imagine circle in R^3.

You can also look at it from the Möbius strip perspective, the Möbius strip being a twisted product (in contrast to the annulus being just the product) of S^1 x R^1, in which case some of the work is already done for you.

The surface.

The surface itself is the Klein bottle, so I would expect its boundary to be some 1-dimensional object, in the same way the boundary of a disk is the circle surrounding it.

A Mobius strip is a twisted non-orientable 2D surface that exists in 3 dimensions, with a 1D edge.

A Klein bottle is a twisted non-orientable 3D volume that exists in 4 dimensions, with a 2D "edge" surface.

You get a Klein bottle from Möbius strip by identifying the boundary 1-cell with itself (plus a twist, whatever). It is not a "3d volume"; there is no 3-cell involved.

From wiki: "Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary (for comparison, a sphere is an orientable surface with no boundary)." (https://en.wikipedia.org/wiki/Klein_bottle)

Then, could you explain what Pxtl meant by “a Klein bottle is a 4d moebius strip”?


One standard way to construct both a Möbius strip and a Klein bottle (and other classic manifolds) is to take a square, and glue some edges together.

For a Möbius strip, you glue together the left and right (say) edges, such that the upper part of one connects to the bottom part of the other (that is, put a twist in the square before you glue).

For a Klein bottle, you additionally glue together the top and bottom edges, but don't twist them. This is what it means to say a Klein bottle is "like a Möbius strip" or "two strips glued together" or similar things.

The "4d" comes in because if you want to do this with a physical object, you need four spatial dimensions unless you're ok with it passing thru itself.

Here's a good picture : http://web.ornl.gov/sci/ortep/topology/topo5.gif The arrows indicate which edges to glue together, and how to line them up.

See also bmm6o's comment here: https://news.ycombinator.com/item?id=11196493


this seems to hinge on a peculiar definition of a knot.

its a mathematical knot, which is essentially definable as a loop of "string" that has crossings that cannot be removed (ie the loop returned to a normal doughnut shape/torus) via any sort of transformation that doesn't involve cutting it.

the trefoil in the article is an example of such a thing.

Most knots we encounter in every day life are not a knot in a mathematical sense, because they can be untied, or don't form a loop.

I'm not familiar with the subject, so I don't quite get that last sentence. It seems to me like most knots I encounter in everyday life is similar to this one: http://www.nationbydesign.com/simple-knot.jpg

This, to me, seems like a knot that cannot be removed via any sort of transformation that doesn't involve cutting it, assuming that the piece of string has no ends and continues forever.

Is my understanding of a knot in the mathematical sense wrong? Or of the involved transformations? If so, can you elaborate?

Your understanding is fairly correct, but it is mathematically easier/nicer to tie the ends together rather than let them run to infinity. That guarantees that you describe exactly how the ends would 'continue forever', and fits better with topology.

See https://en.wikipedia.org/wiki/Knot_theory

Eh, putting a point at infinity (where the ends terminate) is hardly an obstacle to a mathematician. :)

- not all topologies have infinity.

- 'Continue forever' is vague. You will have to specify the direction, as that can affect the knot you get.

Ah. So much simpler, actually! Thanks.

I guess most strings in the real world don't go on forever... And if you have two lose ends, you can untie the knot trivially.

I think that is what your parent comment was getting at.

Take X = all continuous well behaved 1-1 mapping f from [0,1] to R^3, with f(0)=f(1).

A knot is an equivalence class of such functions, equivalence defined as given two functions f and g, you can "morph" one into other continuously - i.e. if you can find a parametrized well-behaved (i.e. continuous, differentiable etc.) function h(x;t) s.t. h(x;0)=f(x) and h(x;1)=g(x) and h(x;t_ is in X for all t.

It's pretty much the definition you'd come up with too.

This summarizes my entire experience with higher mathematics (and in fact academia in general).

To be fair, this is mostly because in order to have an in-depth conversation about a topic you first need to make sure everybody involved shares the same definitions. And in this case it's also because naming things is hard and "regular" knots are not very interesting in mathematics.

Surely a plane can be tied in a knot in R^3 (like a sheet of paper, a blanket) - and therefore trivially so in higher dimensions?

Any such knot could be topologically deformed into a knot made of arbitrarily thin plane segment, which would end up being the same as a line knot. So knots made out of finite plane segments are uninteresting.

I should have been more clear though, I was imagining an infinite 'blanket', which I grab in the 'middle' (not an end) and tie a knot there.

Or does that not count as (mathematical) knotting?

A sheet of paper or a blanket would only be a plane segment.

A string is just a line segment

Yes, but the ends could be extended to infinity in arbitrary directions, so we can tie a knot in a line, not just a segment. You can't do that to the edges of a blanket knot (two sides run into the knot).

A mathematical knot is like a loop of string. There are no ends. Of course, you could cut any mathematical knot wherever you like and make the two ends go to infinity if you'd like.

Thank you, I appreciate the correct definition.

I understand a bit better after that, thanks.

But a knot in a plane segment doesn't need to fold the ends in - I can grab a blanket in the middle, and tie a knot there, then extend edges infinitely still?

I can imagine an infinitely long line with a know in the middle in 3d ... I guess the interesting question is "can you have an infinite plane with a knot in the middle in 5 dimensions?6 dimensions?"

if you selected a single point on a plane and pulled it upwards you could then knot it very similarly to a line couldn't you?

that's like tieing a knot in a loop of string, not like a granny knot ... I think the sort of knot I have in mind is a harder problem

A knot is an embedding of S1 in R3, i.e. a closed loop and not a line segment.

For it to be a mathematical "knot", the edges have to be sewn together into form a knotted sphere. The trivial way to knot a blanket doesn't yield a trivial way to connect all the edges without untying the knot.

A sheet of paper or a blanket is equivalent to D2, the disc, which is not a plane.

Are there any (practical or theoretical) applications for theories of knots in higher dimensional spaces?

> Ordinary knots (that you can tie with string) can only exist in exactly 3 dimensions.

Or a higher dimensional space.

In a 4-dimensional space, 1-dimensional knots are trivially untied, because you can pass strings through strings by "lifting" the string into fourth dimension (strings can "pass through" each other).

Surely you can't be serious.

I am, and don't call me Shirley.

But that's not important right now.

Well I was expecting an article about air traffic control :o

I'm so happy someone beat me to this joke.

Not sure why this is the top comment since this isn't Reddit bla bla. I've had my previous account hellbanned because of upvoting a comment like this. (I can only assume, since I hadn't posted anything in a long time).

I don't think accounts get hellbanned based on voting behavior unless it suggests they are part of a voting ring.

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