I'm not sure whether I have articulated my curiosity well enough here. Please, therefore, bear with me if I need to edit the question, and please forgive me if this is otherwise a nonsense question that cannot be salvaged.
Consider the following Linkin Park logo:
(NB: I'm not sure whether permission is needed to share this logo; if it is needed, let me know and then I'll delete it.)
The logo is one continuous line in the shape of a circle that, if you start at the top end, draws what is nearly a semicircle, before tracing out a stylised $P$, which leads to a stylised $L$, which, in turn, becomes the rest of the circle.
What I have described is a counterclockwise orientation that writes the $P$ and the $L$ in that order - backwards in standard English - and if you wanted to write $L$ then $P$ (by tracing out the logo), then you would need to start at the bottom end (although also counterclockwise).
What type of Mathematics, if any, describes/studies these different orientations of similar images?
Some thoughts:
It might not be graph theory. There is a direction to what is going on here, something an undirected graph doesn't take into account.
It's not knot theory. The logo is not a knot.
It might be a directed graph.
I'm aware of the notion of a winding number, something I encountered in my undergraduate days during my complex analysis and my special functions modules. It seems related.
The question The Mathematics of Symbol Recognition. of mine springs to mind, although I'm not sure why.
We could, I suppose, ignore the perimeter, leaving us with just the $L$ and the $P$, although that would switch top-to-bottom to bottom-to-top, and vice versa.
The order of the $L$ and the $P$ is what I'm interested in most of all here. This is motivated by considering how I would draw the logo.
As you can see, this is a difficult thing to search for online (especially because it's still a little fuzzy to me).
Please help :)
