Möbius Band as Simplicial Complex

Here we join up the top and bottom surfaces, but with a half twist. moebius topology
So we get a one sided surface with a single external boundary line. moebius

In the program FriCAS (described here) there is a function: moebiusBand which generates a minimal triangulation of a Moebius band.

Möbius band: One boundary, shown in red:

(2) -> moebiusBand()

        +1  2  3+
        |2  3  4|
   (2)  |3  4  5|
        |1  4  5|
        +1  2  5+
  Type: FiniteSimplicialComplex(VertexSetAbstract)

moebius band

This diagram can only be realised in 3D, to make it clearer in 2D I have duplicated nodes 3 and 4:

Möbius band

The blue arrows should join up, notice the head of the arrow is at the top on the left and bottom on the right.

 


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flag flag flag flag flag flag Mathematics++: Selected Topics Beyond the Basic Courses (Student Mathematical Library) Kantor, Ida.

Chapters:

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  4. Representations of Finite Groups
  5. Polynomials
  6. Topology

Chapter 6 - Topology. Contains a relatively gentle introduction to homology.

 

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