Cancellative Elliptic Artin Monoids

We define new presentations for elliptic Artin groups. We also show that the elliptic monoids defined by these presentations are cancellative. This solves the failure of cancellativity for the presentations of elliptic Artin monoids that were introduced before in the literature. Our approach also paves the way to construct Garside structures for elliptic Artin monoids and groups.