Prism graphs in tropical plane curves

TL;DR

This paper characterizes when prism graphs can serve as the skeletons of smooth tropical plane curves, establishing a genus limit of 11, and extends previous results on big face graphs.

Contribution

It proves that prism graphs are the skeletons of smooth tropical plane curves if and only if the genus is at most 11, extending Morrison and Tewari's work.

Findings

Prism graphs are valid skeleta for genus ≤ 11.

Prism graphs are not skeleta for genus ≥ 12.

Classification of lattice polygons with specific viewing properties.

Abstract

Any smooth tropical plane curve contains a distinguished trivalent graph called its skeleton. In 2020 Morrison and Tewari proved that the so-called big face graphs cannot be the skeleta of tropical curves for genus $12$ and greater. In this paper we answer an open question they posed to extend their result to the prism graphs, proving that they are the skeleton of a smooth tropical plane curve precisely when the genus is at most $11$. Our main tool is a classification of lattice polygons with two points than can simultaneously view all others, without having any one point that can observe all others.