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Piecewise $$SL_ 2{\mathbb{Z}{}}$$ geometry. (English) Zbl 0768.52009
Pick’s theorem gives a formula for the area of a planar integral polygon (where all vertices lie in $$Z^ 2$$) in terms of the number of integral points in the polygon. The author studies area-preserving maps between such polygons which are linear — i.e. elements of $$\text{SL}(2,Z)$$ — when restricted to each integral triangle of a certain subdivision. Furthermore, to a given polygon $$P$$ he associates a graph $$G(P)$$ whose vertices are integral edges through the interior of $$P$$ (e.g., diagonals of $$P$$) and whose edges correspond to crossings of such. This leads to a simplicial complex $$K(P)$$ where the simplices are subsets of vertices of $$G(P)$$ which are pairwise not joined by edges in $$G(P)$$. The main theorem says that for certain types of integral polygons $$K(P)$$ is a triangulated disk of dimension $$2A+N-2$$, where $$A$$ is the area and $$N$$ is the number of interior integral points.

##### MSC:
 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
##### Keywords:
area-preserving maps; integral polygons
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