Tropical Elliptic Curves

This repository contains the atlas of tropical elliptic curves atlas.pdf as described in our article [??], as well as code and links to datasets we used to generate the atlas.

Datasets

In order to arrive at the atlas we first used mptopcom [2] to compute at the regular full triangulations of the Cayley polytope $C(2\Delta_3, 2\Delta_3)$. From this dataset we filtered the (regular) unimodular triangulations into dataset [4].

On these triangulations we ran the polymake [1] script cycle_length_iso.pl to arrive at the isomorphism classes of graphs that appear among these tropical elliptic curves. The triangulations are numbered and we refer to these numbers as IDs. The resulting data is contained four polymake data files in the folder data of this repository:

• regular_unimodular.representative_graphs: For every ID of a representative of a class of tropical elliptic curves up to graph isomorphism, map this ID to the corresponding graph. The polymake datatype is Map<Int, Set<Array<Int>>>.
• regular_unimodular.representative_triangulations: For an ID as above, map it to the triangulation realizing this graph. The polymake datatype is Map<Int, Array<Set<Int>>>.
• regular_unimodular.cl2ids: The "cl" stands for "cycle length". Map every cycle length to the set of all IDs realizing a tropical elliptic curve with that cycle length. The polymake datatype is Map<Int, Set<Int>>.
• regular_unimodular.classes_histogram: For every ID as in the first two entries, map this ID to the number of regular unimodular triangulations with the same graph. The polymake datatype is Map<Int, Int>

These can then be turned into a large latex document as described below.

Generating latex files

For this switch to the root of this repository and run the following command:

polymake --script scripts/to_latex.pl

(This script has hard coded paths, so it will not work from elsewhere).

Afterwards switch into the latex directory and run the following:

cd latex/
pdflatex main.tex

Note that this will take a while, since this document contains 4009 TikZ pictures.

References

[1] polymake - open source software for research in polyhedral geometry.

[2] mptopcom - computing triangulations of point configurations in parallel.

[3] Regular full triangulations of the Cayley polytope C(2\Delta_3, 2\Delta_3) Dataset on Zenodo.

[4] Regular unimodular triangulations of the Cayley polytope C(2\Delta_3, 2\Delta_3) Dataset on Zenodo.