Can planes be tied in knots in higher dimensions the way lines can in 3D?

http://www.askamathematician.com/2016/02/q-can-planes-sheets-be-tied-in-knots-in-higher-dimensions-the-way-lines-strings-can-be-tied-in-knots-in-3-dimensions/

by joeyespo • 10 years ago

92 51 10 years ago

askamathematician.com

OJFord3• 10 years ago

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?

AgentME• 10 years ago

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

grandalf2• 10 years ago

A string is just a line segment

cynicalkane• 10 years ago

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.

OJFord3• 10 years ago

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?

GFK_of_xmaspast5• 10 years ago

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

gohrt2• 10 years ago

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.

grandalf2• 10 years ago

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

throwaway2048• 10 years ago

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.

anyfoo2• 10 years ago

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?

altrego99• 10 years ago

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.

pluma• 10 years ago

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.

glitcher• 10 years ago

Yes

https://en.wikipedia.org/wiki/File:Airplane_art_1.jpg

cbd1984• 10 years ago

But that's not important right now.

johnhattan• 10 years ago

Surely you can't be serious.

livatlantis• 10 years ago

I am, and don't call me Shirley.

jdblair• 10 years ago

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

huuu• 10 years ago

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

zodiakzz• 10 years ago

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).

pc86• 10 years ago

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

a_c• 10 years ago

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

jordigh• 10 years ago

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.

Pxtl• 10 years ago

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

catnaroek4• 10 years ago

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

bmm6o• 10 years ago

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".

ingenter2• 10 years ago

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

amelius2• 10 years ago

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

Or a higher dimensional space.

ingenter2• 10 years ago

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).

amelius2• 10 years ago

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

Can planes be tied in knots in higher dimensions the way lines can in 3D?

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