Geodesic
Haas' theorem revisited
Épijournal de Géométrie Algébrique, Tome 1 (2017)

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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. 
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     title = {Haas' theorem revisited},
     journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
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Bertrand, Benoît; Brugallé, Erwan; Renaudineau, Arthur. Haas' theorem revisited. Épijournal de Géométrie Algébrique, Tome 1 (2017). doi: 10.46298/epiga.2017.volume1.2030

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