I have a black thread that I wind around the torus along the two directions x and y with arbitrary windings around these directions. As I am winding the thread around the torus, the color of the unwound part of the thread changes from black to white or white to black everytime the thread completes a winding in y direction (note that the color doesn't change for the whole thread but only for the part that hasn't been wound yet). In the end after arbitrary windings in x and y directions (with color changes for y-windings as mentioned) in some order, I return to the original starting point and join the thread there.
I want to know how many topological crossings of black and white parts of the thread do I get after arbitrary number of windings (in a given order) in $x$ and $y$ directions, $n_x$ and $n_y$. By "topological" crossings, I mean, the crossings that I cannot remove w/o changing the winding numbers $n_x$ and $n_y$.
For example, below I represent the torus as a square (that is periodic in the two directions) where the color white has been drawn as a dashed line. Imagine that I start winding the thread at the left lower corner of the square. Then I complete a winding in the x direction and go up in y as I go forward in x. Eventually, I also complete a winding in y. Notice that on completing the winding in the y direction, I change the color from solid (black) to dashed (white). This is an example with $n_x=2$ and $n_y=2$.
So, for $n_x=2$ and $n_y=2$, as you see in the figure, we get one black-white crossing. Is there a right way to approach this problem to find the total number of such black-white crossings for arbitrary number of windings $n_x$ and $n_y$ with a given order of these windings ? I am not hoping for a complete solution but just to know what would be the right tools to use.
PS- in different versions of this problem, I could change the thread color also when winding in $x$ direction is completed. Or I could also change the thread color at some fractions of winding in $x$ or $y$ direction.