Colored admissible tagged edges in a twice-punctured disk model the non-tau-rigid modules of type \widetilde{D}_n and satisfy an intersection-dimension formula.
A geometric model for the module category of a skew-gentle algebra
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In this article, we realize skew-gentle algebras as skew-tiling algebras associated to admissible partial triangulations of punctured marked surfaces. Based on this, we establish a bijection between tagged permissible curves and certain indecomposable modules, interpret the Auslander-Reiten translation via the tagged rotation, and show intersection-dimension formulas. As an application, we classify support $\tau$-tilting modules via maximal collections of non-crossing tagged generalized permissible curves.
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A geometric model for the non-$\tau$-rigid modules of type $\widetilde{D}_n$
Colored admissible tagged edges in a twice-punctured disk model the non-tau-rigid modules of type \widetilde{D}_n and satisfy an intersection-dimension formula.