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Consider the 4 × 4 grid graph:

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Now torify it, i.e. connect its opposing vertices:

enter image description here

How can one tell the difference between a “meridian” and an “equator”?

The difference seems clear when looking at a “continuous” torus embedded in $\mathbb{R}^3$:

meridians

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vs. equators

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In the discrete case there seems to be no difference at all.

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    Yes. I'm pointing out that a topological torus has no intrinsic geometry, and Gerry is pointing out that graphs are topological, not geometric objects.2012-12-04

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The difference between an equator and a meridian is extrinsic to the torus, not intrinsic. A 2D being living on the surface of the torus might be able to distinguish between these two using distance measurements, but from a purely topological point of view, both directions are equal. Only in the embedding into 3D you can describe the difference as whether or not the loop goes around the hole in the torus. As your graph combinatorics does not carry sufficient information for a 3D embedding, you cannot tell the difference from these.