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

TL;DR

This paper introduces a geometric model using homotopy classes of unoriented arcs in a twice punctured disk to describe non-homogeneous tubes in the cluster category of affine type D, extending previous models for rank n-2 tubes.

Contribution

It provides a new geometric framework for understanding non-homogeneous tubes in affine type D cluster categories, especially for rank 2 tubes.

Findings

Model using unoriented arcs in a twice punctured disk

Extension of previous models to rank 2 tubes

Enhanced understanding of non-homogeneous tubes

Abstract

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 $2$.