Title: Presentations of the saturated cluster modular groups of finite mutation type $X_6$ and $X_7$,
Abstract: We give finite presentations of the saturated cluster modular groups of type
$X_6$ and $X_7$. We compute the first homology of these groups and conclude
that they are different from Artin-Tits braid groups and mapping class groups
of surfaces. We verify that the cluster modular group of type $X_7$ is
generated by cluster Dehn twists. Further we discuss several relations between
these cluster modular groups and the mapping class group of an annulus.