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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.

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@Alex, are you still here? Can you accept one of the answers, or indicate in what way they are inadequate? – Gerry Myerson Feb 19 '11 at 22:44
up vote 1 down vote accepted

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.

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I am curious about the same question, but maybe in a simpler example like ABCADEA vs ADEABCA. – Dave Pritchard Feb 17 '11 at 9:47
What happens if we have self avoiding walks? – Alex R. Feb 20 '11 at 3:35

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