Non-Kissing Complexes and Tau-Tilting for Gentle Algebras

We interpret the support τ-tilting complex of any gentle bound quiver as the non-kissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its g-vector fan and prove that it is the normal fan of a non-kissing associahedron.