Haas' theorem revisited

TL;DR

This paper revisits Haas' theorem on patchworking of non-singular plane tropical curves, providing a new interpretation using abstract graphs and topological surfaces to better understand the space of real algebraic curves.

Contribution

It offers a novel topological and graph-theoretic interpretation of Haas' theorem, extending its applicability beyond plane tropical curves to abstract graphs.

Findings

Characterizes the space of patchworkings as an affine subspace of real structures.

Provides a new proof of Haas' original theorem using topological methods.

Connects involutions on surfaces to the algebraic structure of patchworking.

Abstract

Haas' theorem describes all partchworkings of a given non-singular plane tropical curve $C$ giving rise to a maximal real algebraic curve. The space of such patchworkings is naturally a linear subspace $W_C$ of the $\mathbb{Z}/2\mathbb{Z}$-vector space $\overrightarrow \Pi_C$ generated by the bounded edges of $C$, and whose origin is the Harnack patchworking. The aim of this note is to provide an interpretation of affine subspaces of $\overrightarrow \Pi_C $ parallel to $W_C$. To this purpose, we work in the setting of abstract graphs rather than plane tropical curves. We introduce a topological surface $S_\Gamma$ above a trivalent graph $\Gamma$, and consider a suitable affine space $\Pi_\Gamma$ of real structures on $S_\Gamma$ compatible with $\Gamma$. We characterise $W_\Gamma$ as the vector subspace of $\overrightarrow \Pi_\Gamma$ whose associated involutions induce the same action on $H_1(S_\Gamma,\mathbb{Z}/2\mathbb{Z})$. We then deduce from this statement another proof of Haas' original result.