Index of the Kontsevich-Zorich monodromy of origamis in $\mathcal{H}(2)$

Abstract

The Kontsevich-Zorich monodromy of an origami is the image of the action of the Veech group on the non tautological part of the homology. In this paper we make some progress to show, that for origamis in the stratum $\mathcal{H}(2)$ the index of the Kontsevich-Zorich monodromy in $SL_2(\mathbb{Z})$ is either 1 or 3.