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By Nathan Broomhead

Quantity 215, quantity 1011 (second of five numbers).

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This is how it is defined in the physics literature, for example in [27]. In this language, knowing that a is a zig or a zag of a zig-zag path or flow, is equivalent to knowing whether the zig-zag path or flow turns left or right at the head of a. Since there exists a unique zig-zag path or flow containing any given arrow a as a zig (respectively zag), we will refer to this as the zig-zag path or flow generated by the zig (respectively zag) a. In particular, the lift of a single zig or zag of a zig-zag path η to Q, (remembering that it is a zig or zag), generates a zig-zag flow η which projects down to η.

5 it intersects the boundary of f in a zig and a zag. The zag of η is a zig of a uniquely defined zig-zag flow η ∈ X (f ) which crosses η from right to left. 14 this implies that intersection number is strictly positive. Therefore the ray generated by [η ] is at an angle less than π in an anti-clockwise direction from γ. 12. The local zig-zag fan ξ(f ) at the face f of Q, is the complete fan of strongly convex rational polyhedral cones in H1 (Q) whose rays are generated by the homology classes corresponding to zig-zag flows in X (f ).

First we observe that if the lattice polygon is a square of arbitary size, then it is not hard to write down a ‘good’ configuration of curves. For example, if the polygon is the one shown on the left below, then there should be three curves with each of the classes (1, 0), (−1, 0), (0, 1), (0, −1) and these can be arranged as shown on the right. Gulotta’s algorithm transforms a ‘good’ configuration of curves defining a dimer model into another one, in an explicit way that corresponds to removing a triangle from the lattice polygon.

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