Extensions in Jacobian algebras via punctured skein relations

TL;DR

This paper demonstrates how to identify all non-split extensions in Jacobian algebras from punctured surfaces using geometric methods like tagged arcs and skein relations, extending to other surface types.

Contribution

It introduces a geometric approach to compute non-split extensions in Jacobian algebras from punctured surfaces, connecting cluster algebra techniques with surface topology.

Findings

All non-split extensions can be derived from tagged arcs and skein relations.

The geometric interpretation applies to Jacobian algebras from various punctured surfaces.

Examples include type D and other punctured surface cases.

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 form surfaces with punctures. We show examples in type $D$ and in a punctured surface.