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I would like to have a precise definition of the topological space called "full-twist" Möbius strip $M$, i.e. a Möbius band with a $360$ degree twist (the usual Möbius strip has just a half-twist).

This space appear in these questions: (1) and (2). The second one states that $M$ is homeomorphic to a cylinder, which first of all requires some kind of definition for $M$.

I don't think that a parametrization in $\Bbb R^3$ would be a good definition. In general, I've seen the usual Möbius strip defined as a quotient of $[0,1]^2$. Could we do the same in order to define $M$ ? I'm not sure how to do it.

Thank you for your comments!

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Watson
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1 Answers1

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Topologically the full-twist strip and the untwisted strip are identical. The only difference is the way they are immersed in $\Bbb R^3$. So you aren't going to get an intrinsic characterization that is like the quotient space one you mentioned. (To see this, consider how you make a full-twist strip: take the untwisted strip, cut it across, twist it, and then rejoin each point on the cut to exactly the same points it was attached to before the cut. Or imagine twisting the strip in $\Bbb R^4$ instead of in $\Bbb R^3$.)

Knot theory faces a similar problem. Two linked circles are topologically identical to two unlinked circles: they are both the disjoint union of two circles. (Compare these with the boundaries of your two kinds of strips.) But knot theory wants to distinguish not the spaces themselves but the way they are immersed in $\Bbb R^3$. One way this is done is to consider the complement of the two sets in $\Bbb R^3$. Say the full-twist strip is $F_2$ and the no-twist strip is $F_0$. Then consider the spaces $ \Bbb R^3-F_2$ and $\Bbb R^3-F_0$. These are topologically distinct. But to really carry this out may be difficult.

MJD
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  • Is there a way to visualize the untwisting of a "full-twist band" in $R^4$? How to convince myself visually that this can be done? – Alex Jul 29 '23 at 00:58
  • First, try to imagine turning two linked circles into two unlinked circles in $R^4$. Once you can do that, consider that the boundary of the untwisted band is two unlinked circles, and the boundary of the full-twist band is two linked circles. In each case you can think of the strip as a whole family of parallel circles. Perform a transformation like the one that unlinked two circles, except on all the circles at once. – MJD Jul 29 '23 at 02:16
  • I have another way in mind, which is easier to visualize, but I don't know how to describe it. – MJD Jul 29 '23 at 02:17
  • Wow this actually works. Thank you. Any hints on the other easier way? – Alex Jul 29 '23 at 06:56