, The double G D is obtained by taking the diagonals of the quad-graph G and adding a white vertex at the crossing of the diagonals. It is bipartite and has two kinds of black vertices corresponding to vertices of G and vertices of G * . The associated quad-graph (G D ) is obtained by dividing the quadrangular faces of G into four, see Figure 16. The graph G Q is the dual graph of the superimposition of the quad-graph G and of the double graph G D . It has three kinds of faces containing in their interior either a vertex of G, a vertex of G * , or a white vertex of G D ; the latter are quadrangles whose pair of parallel edges correspond to primal and dual edges of G. The quad-graph (G Q ) is that of G D with an additional quadrangular face for each edge of G , which should be thought of as "flat", see Figure 15 and the discussion below. Each train-track of G induces two train-tracks that are anti-parallel in G D and make antiparallel bigons in G Q , see Figures 16 and 15. Therefore, if G is an isoradial graph, i.e., if its train-tracks do not self-intersect and two train-tracks never intersect more than once, then the associated bipartite graphs G D and G Q are isoradial and minimal, respectively. Moreover, an isoradial embedding of G naturally induces an isoradial embedding of G D and a minimal immersion of G Q , as follows. As proved by Kenyon-Schlenker [KS05], an isoradial embedding is given by some half-angle map ? on the oriented train-tracks of G, such that the half-angles associated to any given oriented train-track and to the same train-track with the opposite orientation differ by ? 2 . One can then simply define the induced half-angle maps ? D and ? Q by associating to each oriented train-track of G D and G Q the half-angle of the unique oriented train-track of G it is parallel to. This is illustrated in Figures 16 and 15. If ? defines an isoradial embedding of G
, Connection to the dimer model on the graph G Q arising from the Ising model Consider a half-angle map ? Q ? X G Q as above. Following Section 2.3, let us compute the discrete Abel map on vertices of the quad-graph (G Q ) , taking as reference point a vertex v 0 of G
using the notation of Figure 15, and recalling that ? is defined as an element of R/?Z, we have the following equalities modulo ? ? v ? V, ?(v) = 0, ? f ? V * , ?(f) = ? 2 , ?(b 1 ) = ? + ? 2 , ?(b 2 ) = ?, ? ,
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