
Can planes be tied in knots in higher dimensions the way lines can in 3D? - joeyespo
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/
======
ingenter
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](http://i.imgur.com/2rIKJ8E.png)

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

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

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

~~~
startling
The surface.

~~~
catnaroek
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.

~~~
Manishearth
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.

~~~
GFK_of_xmaspast
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.

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

~~~
throwaway2048
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.

~~~
anyfoo
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](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?

~~~
Someone
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](https://en.wikipedia.org/wiki/Knot_theory)

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

~~~
Someone
\- not all topologies have infinity.

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

------
OJFord
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?

~~~
cynicalkane
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.

~~~
OJFord
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?

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

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

Or a higher dimensional space.

~~~
ingenter
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).

------
glitcher
Yes

[https://en.wikipedia.org/wiki/File:Airplane_art_1.jpg](https://en.wikipedia.org/wiki/File:Airplane_art_1.jpg)

~~~
johnhattan
Surely you can't be serious.

~~~
livatlantis
I am, and don't call me Shirley.

