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Comment by a_c | Hacker News Reader

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”?

Sure.

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

Thanks!

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