Haas' theorem revisited

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.

22 pages, 14 figures

References in corpus (3)

• Enumerative tropical algebraic geometry in R2
• Brief introduction to tropical geometry
• Tropical Homology

Cited by in corpus (2)

• Real phase structures on matroid fans and matroid orientations
• Hyperbolic plane curves near the non-singular tropical limit