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

TL;DR

This paper investigates the index of the Kontsevich-Zorich monodromy for origamis in the stratum (2), showing it is either 1 or 3, thus advancing understanding of their monodromy properties.

Contribution

It provides new results on the possible indices of the Kontsevich-Zorich monodromy in (2) origamis, specifically narrowing it down to 1 or 3.

Findings

The index of the monodromy is either 1 or 3 for origamis in (2).

Progress in understanding the monodromy's image in SL_2().

Enhanced classification of origamis based on monodromy index.

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.