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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.

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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