Persistence and Stability of the $\mathbb{A}_n$ Quiver

We introduce two new distances for zigzag persistence modules. The first uses Auslander-Reiten quiver theory, and the second is an extension of the classical interleaving distance. Both are defined over completely general orientations of the $\mathbb{A}_n$ quiver. We compare the first distance to the block distance introduced by M. Botnan and M. Lesnick and obtain the full set of sharp Lipschitz bounds between the two (as bottleneck distances) over pure zigzag orientations. The final portion of the paper presents sharp Lipschitz bounds necessary for the extended interleaving distance to dominate the distance that is created from the Auslander-Reiten quiver. These bounds are obtained for general orientations of the $\mathbb{A}_n$ quiver.