Tropical elliptic curves in 3-space

TL;DR

This paper classifies tropical elliptic curves in 3-space by analyzing trivalent graphs from intersecting quadratic surfaces, using extensive combinatorial triangulations to understand their structure.

Contribution

It provides a comprehensive classification of tropical elliptic curves as intersections of quadrics in 3-space, based on a large-scale combinatorial enumeration.

Findings

Identified 4,009 distinct tropical elliptic curve graphs

Connected these graphs to regular unimodular triangulations of a 4D polytope

Established a classification framework for tropical elliptic curves in 3-space

Abstract

We classify trivalent graphs with 16 vertices and 16 edges that arise from intersecting two quadratic surfaces in tropical 3-space. There are 4,009 such graphs, representing maximally degenerate stable models of elliptic curves realized as tropical complete intersections of two quadrics. Our classification is derived from 405,246,030 regular unimodular triangulations of the 4-dimensional Cayley polytope.