Tropical hyperelliptic curves in the plane

TL;DR

This paper characterizes tropical hyperelliptic curves as certain planar embedded graphs derived from polygon triangulations and identifies combinatorial obstructions preventing some graphs from tropical embedding.

Contribution

It proves hyperelliptic graphs in the tropical setting originate only from specific polygon triangulations with collinear interior points.

Findings

Hyperelliptic graphs correspond to triangulations with collinear interior points.

Certain graphs cannot be embedded tropically in the plane due to combinatorial obstructions.

Hyperelliptic graphs are characterized by their origin from specific polygon triangulations.

Abstract

Abstractly, tropical hyperelliptic curves are metric graphs that admit a two-to-one harmonic morphism to a tree. They also appear as embedded tropical curves in the plane arising from triangulations of polygons with all interior lattice points collinear. We prove that hyperelliptic graphs can only arise from such polygons. Along the way we will prove certain graphs do not embed tropically in the plane due to entirely combinatorial obstructions, regardless of whether their metric is actually hyperelliptic.