Torus as Simplicial Complex

We can represent a torus as a delta complex like this:

That is: there is one vertex 'x' and two independent loops on it.

To avoid having multiple copies of the vertex 'x' we can embed it in 3 dimensions, then first wrap the loop 'n' back to itself: torus
We can then loop this whole loop back to itself: torus

In the program FriCAS (described here) there is a function: torusSurface which generates a minimal triangulation of the surface of an 3-dimensional torus.

Torus -

(1) -> torusSurface()

        +1  2  3+
        |2  3  5|
        |2  4  5|
        |2  4  7|
        |1  2  6|
        |2  6  7|
   (1)  |3  4  6|
        |3  5  6|
        |3  4  7|
        |1  3  7|
        |1  4  5|
        |1  4  6|
        |5  6  7|
        +1  5  7+
  Type: FiniteSimplicialComplex(VertexSetAbstract)
    

No boundaries, the red arrows at the top and bottom join up and the blue arrows on the left and right join up.

torus

 


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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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