Hello Bruce,
> I am now reading Chapter I of Lando and Zvonkin so I
> have a much better idea of your interests.
Cool! But beyond ramified covers of the sphere (as in [LZ]) and
Strebel differentials (as in the paper [MP] mentioned by Tom), I used
ribbon graph as a topological object : a graph embedded in a surface.
In particular they define a simplicial structure from which we may
define a complex with a derivation which gives homology groups of the
surface (as well as Poincare duality). The latter is the main part of
what I have implemented and from which I'm able to compute what is
called the spin-parity of a translation surface (see [KZ]) which
depends only on some angle data between each half-edge.
> I allow boundary points which I have not seen;
> but you did mention partial permutations. I will
> have a look at your code and see if I can figure
> out what you have implemented.
In a sense I do not allow boundary point, but it depends on what you
mean by boundary point. Every permutation I consider are defined on a
subset of {0,1,...,n} and I call it partial if the domain (which
equals the range) is not {0,1,...,n}. The terminology I used is
perhaps misleading... In your definition, a boundary point correspond
a half-edge which is not glued to another half-edge, isn't it ? But,
you can think of it as a fixed point for the involution defining the
edges. My question is does the other two permutations (the vertex
permutation and face permutation) are well defined on these boundary
points ? In other words, does these half-edges are embedded in the
surface ?
[LZ] Lando, Zvonkine, "Graphs on Surfaces and Their Applications", Springer
[MP] Mulase, Penkava, http://arxiv.org/abs/math-ph/9811024
[KZ] Kontsevich,Zorich, http://arxiv.org/abs/math/0201292
Best,
Vincent
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