A note on consistency conditions on dimer models

Abstract

We show that for a non-degenerate dimer model, both the first consistency condition of Mozgovoy and Reineke and the properly-orderedness condition of Gulotta are equivalent to a condition on zigzag paths, which goes back to Hanany and Vegh. The last condition is used in arXiv:0905.0059 to study the behavior of a dimer model under the operation of removing a vertex from the lattice polygon and taking the convex hull of the rest.

17 pages, 23 figures; an expanded version of Section 5 of arXiv:0905.0059, to appear in RIMS Kokyuroku Bessatsu