One can define two winding numbers for a 1D path on a 2-torus. They can be easily visualized on the "doughnut" representation (cf this image):
A 2-torus can also be visualized as a square with periodic conditions, and we can define the same windings as how many times a curve crosses the horizontal or vertical boundaries (cf this image):
I am now wondering how one can generalize this to a "winding" number for a 2-torus in a 3-torus. We can start by visualizing the 3-torus as a cube, now identifying opposite edges as the same.
Then I can think of three ways different ways of embedding a 2-torus in this 3-torus, cf this sketch:
All the 2-torus in blue seem topologically equivalent when considered alone, but I guess that they are not when embedded in a 3-torus?
Are you aware of a proper way to classify these 2-torus inside 3-torus?
Some starting point may be that the "doughnut" like on the left does not reach the boundaries of the 3-torus, where the "cylinder" crosses it once and the "plane" crosses it twice?
But then what about a "tilted cylinder" like this?:
It crosses two boundaries but seems different than a "plane".
I know this is not very rigorous, but any help or direction would be very much appreciated!