Combinatorial spin structures I would like to know how to define spin structures combinatorially, for an oriented smooth manifold equipped with a triangulation. In the case of a 2d manifold, spin structures correspond to equivalence classes of Kasteleyn orientations (i.e. orientations of edges so that every face has an odd number of clockwise oriented edges). This fact is important in the theory of dimer models. 
In general (for arbitrary dimensions) some sort of construction is proposed in http://arxiv.org/pdf/1306.4841.pdf   but it is horribly complicated, I do not even see how it is related to Kasteleyn orientations in the 2d case. I would be quite happy to see a combinatorial description of spin structures for 3d and 4d triangulations.
Upd: there is an obvious generalization of the 2d definition to n dimensions. One could assign orientations to (n-1)-dimensional simplices so that for a given n-simplex an odd number of its faces have the orientation opposite to that of the n-simplex. Does this do the job, or is it wrong for some reason?
 A: Here's the standard answer, but perhaps you are looking for something different?
A spin structure on a manifold $M$ is determined by a framing of the tangent bundle $TM$, restricted to the 1-skeleton.  (i.e. the framing is only defined on the 1-skeleton.)  This framing is required to be the bounding framing on the boundary of each 2-cell (i.e. the framing can be extended to the 2-skeleton).  Two such 1-skeleton framings determine the same spin structure if and only if they are homotopic.  Such homotopies are generated by local moves which change the framing at a 0-cell and at all of the 1-cells adjacent to that 0-cell.
Kirby and Taylor have a nice paper on low-dimensional Spin and Pin structures -- http://www3.nd.edu/~taylor/papers/PSKT.pdf

[Added later]
(1) As Ryan points out in a comment, making the above idea explicit enough to be implemented on a computer is the point of his paper (which is linked to in the original question).
(2) If you are willing switch to the dual space -- functions on spin structures rather than spin structures themselves -- then there is the following relatively simple description which works in any dimension.  Define a ribbon in $M$ be be a 1-dimensional submanifold $S \subset M$ equipped with a framing of $TM|_S$.  Consider (finite linear combinations of) isotopy classes of ribbons in $M$ modulo the following three moves.


*

*If $S$ and $S'$ differ by a framed saddle move, then $S \sim -S'$.

*If $S$ and $S'$ differ by adding a single "kink" or "twist" to the framing, then $S \sim -S'$.

*If $S$ and $S'$ differ by adding/removing a small unknotted circle with standard (non-bounding) framing, then $S \sim -S'$.


One can show that the above vector space is canonically isomorphic to the vector space of functions on (equivalence classes of) spin structures on $M$.
Note that if the three occurrences of $S \sim -S'$ above are changed to $S \sim +S'$, then the vector space is canonically isomorphic to finite linear combinations of elements of $H_1(M; \mathbb Z/2)$, which in turn is isomorphic to functions on $H^1(M; \mathbb Z/2)$.  This is the counterpart to the fact that the set of spin structures is a torser for $H^1(M; \mathbb Z/2)$.
The above vector space naturally extends to a (fully extended) TQFT.
