EXTENSIONS IN JACOBIAN ALGEBRAS VIA PUNCTURED SKEIN RELATIONS

Abstract

Given a Jacobian algebra arising from the punctured disk, we show that all non-split extensions can be found using the tagged arcs and skein relations previously developed in cluster algebras theory. Our geometric interpretation can be used to find non-split extensions over other Jacobian algebras arising from surfaces with punctures. We show examples in type  and in other punctured surfaces.