Prism graphs in tropical plane curves

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.