Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls

The Milnor fibre of a Q-Gorenstein smoothing of a Wahl singularity is a rational homology ball B_{p,q}. For a canonically polarised surface of general type X, it is known that there are bounds on the number p for which B_{p,q} admits a symplectic embedding into X. In this paper, we give a recipe to construct unbounded sequences of symplectically embedded B_{p,q} into surfaces of general type equipped with non-canonical symplectic forms. Ultimately, these symplectic embeddings come from Mori's theory of flips, but we give an interpretation in terms of almost toric structures and mutations of polygons. The key point is that a flip of surfaces, as studied by Hacking, Tevelev and Urzúa, can be formulated as a combination of mutations of an almost toric structure and deformation of the symplectic form