Find a manifold which contains embedding of $K_5$

Question

$K_5$ graph is not planar . I was asked to find a manifold which contains embedding of $K_5$ and use $5$ squares to represent $K_5$ "on" my new manifold.

Embedding means that it can be drawn on the manifold in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross.

I can find embedding of $K_5$ in Three-dimensional space but this is not a $5$ squares representation ..

I don't know what I am looking for, so I'd be glad if anyone could help.

p.s This is my first question here so please be patient

thanks, bar

Answer 1

$K_5$ can be embedded into a torus.

Answer 2

Well, I can only guess what "5 squares representation" means, but this is an embedding of $K_5$ into a torus where several squares are kind of "visible" (although they are not faces of this toroidal embedding of $K_5$ - I am not sure what is the correct English terminology for this).

(I have made a picture using metapost. It is figure 5 in the metapost source code, which I have put in pastebin here.)

You can probably find many ways how K5 can be embedded into toroid if you simply google for k5 toroid. You might have a look of some of these embeddings - it might help your intuition about this problem and maybe you will something which is interesting for you in the connection with the problem you are trying to solve.)

EDIT: My original picture contained some edges which were not supposed to be there, so I have replaced it with a new one.