Convex lattice polygons with all lattice points visible

Abstract

Two lattice points are visible to one another if there exist no other lattice points on the line segment connecting them. In this paper we study convex lattice polygons that contain a lattice point such that all other lattice points in the polygon are visible from it. We completely classify such polygons, showing that there are finitely many of lattice width greater than 2 and computationally enumerating them. As an application of this classification, we prove new obstructions to graphs arising as skeleta of tropical plane curves.

Prism graphs in tropical plane curves

“…Now we find those polygons with lattice diameter 2 and lattice width 3. As argued in the proof of [11,Proposition A.1], such a polygon P must have one of the first two polygons in Figure 6 as P int , and due to its lattice width must be a subpolygon of one of the polygons appearing in Figure 8. For each of these two polygons, we run through a series of arguments to find all lattice diameter-2 subpolygons, up to equivalence.…”

“…We say two lattice points p and q are visible to one another if the line segment pq contains no other lattice points besides p and q. Following [11], a panoptigon is a polygon P such that every lattice point q ∈ P ∩ Z 2 is visible from some fixed p ∈ P ∩ Z 2 .…”

“…First assume lw(P ) ≤ 2. All lattice polygons of lattice width at most 2 are classified in [11,Theorem 2.3], a result modeled off a similar classifications found in [8] and [4]. Polygons of lattice width 1 and lattice diameter at most 2 are (possibly degenerate) trapezoids of height 1 with at most 3 lattice points at each height; and those of lattice width 2 are hyperelliptic polygons of genus 2, since those of higher genus have Figure 7: Polygons of lattice diameter 2 and lattice width at most 2 at least 4 lattice points at some common height.…”

“…One recent contribution to this family of results came in [11], which studied the so-called bigface graphs through the lens of tropical planarity. A planar graph G is called a big-face graph if for every planar embedding of G, at least one bounded face (the "big face") shares an edge with every other bounded face.…”

“…For the "only if" direction, we must consider the two embeddings of the prism graph: the standard and the non-standard embedding. The standard embedding has a big-face, and so could only arise from a polygon P whose interior polygon is a panoptigon; by [11] it follows that such a polygon P (and thus the prism) has genus at most 11.…”