EXTENSIONS IN JACOBIAN ALGEBRAS VIA PUNCTURED SKEIN RELATIONS SALOM ´ ON DOM ´ INGUEZ - ANA GARC ´ IA ELSENER Dedicated to the memory of Andrzej Skowro´ nski. 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. 1. Introduction Jacobian algebras arising from surfaces were defined by Labardini-Fragoso in [LF09] building in the works [DWZ08] and [FST08]. The reader can also find an overview of Labardini’s work on Jacobian algebras here [LF16]. Given a compact Riemann surface with some disk removed, in order to create boundaries, and adding marked points in each boundary component and in the interior of the surface, we can define a triangulation. A triangulation is a finite set of curves (up to isotopy) splitting the surface into (ideal) triangles. A marked point in the interior of the surface is called a puncture. The simplest punctured surface is the once-punctured disk. This surface is associated to the cluster category of type D, see [Sch08]. A triangulation of the punctured disk defines a cluster-tilted algebra of type D [BMR + 06, BBMR07]. More generally, a punctured surface defines a cluster category [QZ17], and a triangulation of the surface defines a Jacobian algebra in the sense of [LF09]. Cluster categories associated to unpunctured surfaces were defined by Br¨ ustle-Zhang [BZ11]. For the unpunctured version, Jacobian algebras arising from surface triangulations are gentle algebras and were studied by Assem et. all in [ABCJP10], where string modules are defined by arcs that do not belong to the triangulation. Gentle algebras and their module categories are very well understood. Building on these works, and on their knowledge on gentle algebras and cluster algebras arising from unpunctured surfaces, Canakci and Schroll [CS17] study non-split extensions in module categories of gentle algebras arising from surface triangulations. They show how to find non-split extensions and, as a consequence, they find non- split triangles in the cluster category. The non-split extensions and triangles arise from skein relations over the cluster algebra, this is viewed as a geometric operation that smooths arc crossings in the interior of the surface. See in the next figure the pair of arcs α and β crossing. In our convention α + β means walk along α and turn right following β before reaching the crossing point, and α − β is the same but turning left. α β α + β α − β Skein relations were used to study bases of cluster algebras arising from surfaces (and to prove the Fomin- Zelevinsky positivity conjecture over said algebras). See [MW13, MSW11]. In the trivial coefficients case, skein relations are well understood, and formulas on cluster algebras arising from surfaces with punctures are known. See in the next figure the rule α + β and α − β in the punctured setting. Note that in one case we see a self-intersection. We use cluster algebra equations to find a different presentation. In this work we use skein relations for cluster algebras over a once punctured surface to find all non-split extensions over a cluster tilted algebra of type D. We use these extensions to find triangles in the cluster category and then we see when these triangles induce non-split extensions over Jacobian algebras. 1 arXiv:2108.07844v1 [math.RT] 17 Aug 2021