Let $A = \mathcal{k}Q/I$ be a gentle algebra (where $\mathcal{k}$ is algebraically closed). In the paper [Auslander-Reiten Sequences with Few Middle Terms and Application to String Aglebras](https://pub.uni-bielefeld.de/download/1776051/2312059/Ringel_068.pd), Butler and Ringel show that string and band modules classify the indecomposable modules of $A$ ([pages 157–161](https://pub.uni-bielefeld.de/download/1776051/2312059/Ringel_068.pdf#page=13)). To flesh out the details a little more, for each string $c$ of $Q$ they produce a *string module* $M(c)$. And for each cyclic string $b$ they produce a family of *band modules* $M(b,x,n)$ where $x \in \mathcal{k}^*$ and $n \geq 1$.

I am trying to compare this to the classification of indecomposable representations of the $2$-Kronecker quiver. But as an example I don't see where the indecomposable representation 

$$\mathcal{k}\overset{0}{\underset{1}{\rightrightarrows}} \mathcal{k}$$ 

appears in Butler and Ringel's classification. What am I missing?