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It is known that the torus can be cut into a Möbius strip with an even number $n$ of half twists(half twist means rotation 180 degree). I am asking if it is possible to $n$ to be zero?

user113715
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1 Answers1

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If you think of a torus by taking a cylinder and identifying its ends, then it is clear that cutting along this circle gives you a cylinder (a Möbius strip with zero half twists).

David Kleiman
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  • Thank you, I got the idea. I just have a small question whether a strip with even number of half twists is equivalent to annulus (strip with no half twists) – user113715 Feb 01 '16 at 07:11
  • @user134875 Topologically, yes. If you think a Möbius strip as a unit square with the ends identified, then you can see that they are the same. – David Kleiman Feb 01 '16 at 07:27
  • @user134875 see the answer here for a more detailed explanation http://math.stackexchange.com/questions/1379300/topologically-distinguishing-mobius-strips-based-on-the-number-of-half-twists – David Kleiman Feb 01 '16 at 07:29
  • I understand, I am very grateful. Thank you. – user113715 Feb 01 '16 at 16:10