Directed arcs on a surface

2011-06-24T20:39:45Z

This question is a little odd. I have specific class of structures on a surface, which satisfy several nice properties, and I want to know if they are more natural geometric structures in disguise (my guess is a spin structure with some extra information).

Let $S$ be a connected, oriented surface (possibly with boundary $\partial S$). Let $M$ be a finite collection of marked points, such that every component of $\partial S$ has at least one marked point. Define an arc in $S$ be an smooth, unoriented curve in $S$ with endpoints in $M$; considered up to endpoint-fixing isotopy (we prohibit contractible paths).

I now introduce an odd-seeming structure on $S$, which I believe corresponds to some intrinsic geometric information (but I don't know what). A directing of $S$ assigns an orientation to each arc in $S$; equivalently, it chooses an endpoint to be the 'source'. A directing is clockwise if, for any three arcs $a,b,c$ which bound a triangle in $S$, an odd number of $a,b,c$ are oriented clockwise around the interior.

Example. Consider the disc with $n$ marked points, all on the boundary. Starting with any marked point and moving counter-clockwise, number the marked points $1$ through $n$. Then there is a clockwise directing such that an arc goes from $p$ to $q$ whenever $p < q$.

Clockwise directings satisfy several remarkably nice properties.

If $\partial S\neq \emptyset$, clockwise directings always exist. (The condition on the boundary cannot be removed)
A clockwise directing is completely determined by its restriction to the arcs in a triangulation of $S$. In fact, any choice of orientation of the arcs in a triangulation which satisfies the 'clockwise' condition determines a unique clockwise directing.
There are finitely many clockwise directings.
Any two clockwise directings may be related by a sequence of elementary moves. (1) For a given marked point $p$, reversing the orientation of each arc which has one endpoint at $p$. (2) for a given loop $l$ in $S$, reversing the orientation of each arc which intersects $l$ an odd number of times.

For these reasons and others, I believe that this contrived structure is equivalent to a more natural geometric structure. Thus, my question is,

Is there a class of geometric structures on $S$ such that a choice of clockwise directing is equivalent to a choice of such a structure?

Specifically, I suspect that this corresponds to an extended version of a spin-structure. That is, I conjecture that a clockwise directing on $S$ is equivalent to a choice of spin-structure on $S$, together with some extra information at the marked points. However, I know very little about spin structures, and so I have no idea if there is a natural way to add extra information at a marked point.

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Directed arcs on a surface