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Moving out the edges of a lattice polygon. (English) Zbl 1237.52002
Summary: We review previous work of (mainly) Koelman, Haase and Schicho, and Poonen and Rodriguez-Villegas on the dual operations of (i) taking the interior hull and (ii) moving out the edges of a two-dimensional lattice polygon. We show how the latter operation naturally gives rise to an algorithm for enumerating lattice polygons by their genus. We then report on an implementation of this algorithm, by means of which we produce the list of all lattice polygons (up to equivalence) whose genus is contained in \(\{1,\dots ,30\}\). In particular, we obtain the number of inequivalent lattice polygons for each of these genera. As a byproduct, we prove that the minimal possible genus for a lattice 15-gon is 45.

MSC:
52A10 Convex sets in \(2\) dimensions (including convex curves)
52C05 Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry)
Software:
Magma
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