Question

Let $G=(V,E)$ be an isoradial graph. In other words the graph can be imbedded into the plane such that each face (plaquette) can be circumscribed a circle of radius 1 with the circle's center belonging to the closure of the face. Let $Z$ be the set of circle centers. Then for any walk $C$ on $G$ that starts at some set point $a$, let $n(C,z)$ be the winding number of $C$ around $z\in Z$. Note that my definition of winding number here is not restricted to being an integer and corresponds to the arc traced out by a curve relative to $z$. The question is whether or not specifying winding numbers at each $z$ corresponds to a unique $C$? Moreover, is there a consistency check which makes sure that a given configuration of winding numbers corresponds to some walk?

Note: isoradiality may be unnecessary in this assumption, but my intent is to work with something nice looking. I would be fine with answers restricted to say, a square lattice.

Answer 1

Are two walks considered to be the same if they use the same edges but in a different order? If not, walks ABCDAECFA and ABCFAECDA could have the same winding number around every point in the plane, yet they'd be different walks.

EDIT: And what if your graph is a tree? Then the set of centers is empty, and the walks you want exist, vacuously.