A geometric model for the non-homogeneous tubes of the cluster category of affine type D

In this article, we give a geometric model for non-homogeneous tubes of the cluster category of the affine type $D$. This model is given in terms of homotopy classes of unoriented arcs in the twice punctured disk. In particular, we extend the geometric model for the tube of rank $n-2$ given in arXiv:2407.11232 to the two tubes of rank $2$.