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1. Some background knowledge

Definition. A torus, informally, is the doughnut-shaped surface that we get by taking a square made out of some arbitrarily-stretchy material and gluing together opposite sides. enter image description here More formally, we can think of the torus as the collection of all points $(x, y) ∈ R^2 $under the equivalence relation $(x, y) ∼ (a, b)$ whenever $x − a, y − b ∈ Z$. In other words, this is simply taking the square $[0, 1] × [0, 1]$, “gluing” the edge $\{0\} × [0, 1]$ to the edge ${1} × [0, 1]$, and finally gluing the edge $[0, 1] × \{0\}$ to th edge $[0, 1] × \{1\}$.

We draw $K_5$, $K_{3,3}$, and $K_7$ below to give three such examples of such graphs: enter image description here

2. Confusion in the paper

By using above fundamental polygon of the torus. I can understand this.

But when I read this article:The Toroidal Crossing Number of $K_{m,n}$ I was very surprised by its abnormal torus drawing.

For example:$K_{3,12}$, $K_{4,8}$, $K_{6,6}$,

enter image description here enter image description here enter image description here

They don’t seem to be used normally fundamental polygon of the torus. I can’t tell if they are drawn on the torus. If so, how to transform it into a drawing method by using fundamental polygon of the torus. If it’s difficult, I would like to find a way to judge if these graphs are drawn on the torus.

Any suggestions are precious!

The article link is attached here The Toroidal Crossing Number of $K_{m,n}$

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    $\begingroup$ It is just a slightly different presentation of the torus as a quotient of a rectangle. For example consider Figure 1. Imagine cutting the picture along the line from vertex [3] at bottom left to the vertex [3] at the top. Swap the left and right pieces and glue them back along the line [1]-[3]. Now you have the standard presentation. $\endgroup$
    – Gregory Arone
    Commented Mar 10, 2021 at 7:06
  • $\begingroup$ Thank you very much, it looks nice! $\endgroup$
    – Licheng Zhang
    Commented Mar 10, 2021 at 7:34

1 Answer 1

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You still have the fundamental "square", but it is slanted:

enter image description here

You can move that one bit over:

enter image description here

If you shear it back, you have a fundamental square.

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  • $\begingroup$ Nice! Thank you so much, I understand it this way $\endgroup$
    – Licheng Zhang
    Commented Mar 10, 2021 at 7:35

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