Okay, I have defined a number of variables by the notation c\s, where \s is a number in a foreach loop. When I refer to that later on, things like c0 and c1 seem to work just fine.
But I'm not naming a path L\s and the code seems to work, but when I tell TikZ to draw that path, it just returns a blank.
My guess is that \L1 is not what this path is called?
Welcome to the wonderful world of TeX macro expansion! The problem is on several levels. First, \L is already defined as a macro, so that may be confounding the problem. Second, \L1 is NOT a valid TeX macro: it is interpreted as \L 1. You CAN define a macro whose name is L1, but you would access it via \csname L1\endcsname. This leads us to another problem: from what I can tell, TikZ uses the command you enter as an argument to \let somewhere, which means \csname is being interpreted as the new name. I have tried a combination of \expandafter and \noexpand with no success, but I would strongly advise you to do this another way.
I may keep tinkering with this, but any solution I come up with will probably appear convoluted and/or "high-level" in nature.