# indecomposable modules of gentle algebras

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 Algebras, Butler and Ringel show that string and band modules classify the indecomposable modules of $$A$$ (pages 157–161). 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?

If the $$2$$-Kronecker quiver is
$$\overset 1\circ\overset{\alpha}{\underset{\beta}{\rightrightarrows}} \overset 2\circ ,$$
$$\mathcal{k}\overset{0}{\underset{1}{\rightrightarrows}} \mathcal{k}$$
corresponds to the string module $$M(\beta)$$.