Dessins, their delta-matroids and partial duals

Abstract

Given a map $\mathcal{M}$ on a connected and closed orientable surface, the delta-matroid of $\mathcal{M}$ is a combinatorial object associated to $\mathcal{M}$ which captures some topological information of the embedding. We explore how delta-matroids associated to dessins d'enfants behave under the action of the absolute Galois group. Twists of delta-matroids are considered as well; they correspond to the recently introduced operation of partial duality of maps. Furthermore, we prove that every map has a partial dual defined over its field of moduli. A relationship between dessins, partial duals and tropical curves arising from the cartography groups of dessins is observed as well.

arXiv comment34 pages, 20 figures. Accepted for publication in the SIGMAP14 Conference Proceedings