Cluster categories for marked surfaces: punctured case

We study the cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of $\operatorname{Ext}^1$ as intersection numbers and the Auslander-Reiten translation as tagged rotation. An important consequence is that the cluster exchange graphs in such cases are connected (which is not true for some closed surfaces).