Here's a page from "Knotted Surfaces and Their Diagrams" by J. Scott Carter: http://i.imgur.com/2rIKJ8E.png
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.
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.
A Klein bottle is a twisted non-orientable 3D volume that exists in 4 dimensions, with a 2D "edge" surface.
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
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.
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?
- 'Continue forever' is vague. You will have to specify the direction, as that can affect the knot you get.
I think that is what your parent comment was getting at.
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.
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.
Or does that not count as (mathematical) knotting?
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?
Or a higher dimensional space.