# Directed arcs on a surface

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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This looks quite similar to the concept of Kasteleyn orientations, which are in bijection with spin structures on the surface. But I can't seem to write down a meaningful correspondence yet... –  Gjergji Zaimi Jun 24 '11 at 21:05
Gjergji - Thanks for the tip on Kasteleyn orientations! I'm not sure I have fully digested their definition, but I believe my 'clockwise directings' are the same as Kasteleyn orientations of trivalent graphs on surfaces, with a leaf ending on every component of the complement of the marked points in $\partial S$. To see this, take a triangulation, and consider the dual trivalent graph. The orientations of the triangulation gives an orientation of the graph. –  Greg Muller Jun 25 '11 at 0:55
However, allowing the graph to end on the boundary means that these correspond to more than just a spin structure. For example, the disc with n marked points on the boundary has $2^{n-1}$ clockwise directings, but only 1 spin structure. –  Greg Muller Jun 25 '11 at 0:56
There is a relevant article: "Dimers on surface graphs and spin structures, I and II" by Cimasoni and Reshetikhin (on the arxiv). The second part seems to study the case of surfaces with boundary. It seems that their definition of Kasteleyn orientations is close to yours (when restricted to triangles as you said). Also presumably some trivial cases have to be excluded (like three points on a sphere: which triangle to consider? ) –  Sylvain Bonnot Jun 25 '11 at 21:13