Rigidity of the saddle connection complex

For a half-translation surface $(S,q)$, the associated saddle connection complex $A(S,q)$ is the simplicial complex where vertices are the saddle connections on $(S,q)$, with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. We prove that any simplicial isomorphism $\varphi :A(S,q)\to A(S^{\prime },q^{\prime })$ between saddle connection complexes is induced by an affine diffeomorphism $F:(S,q)\to (S^{\prime },q^{\prime })$. In particular, this shows that the saddle connection complex is a complete invariant of affine equivalence classes of half-translation surfaces. Throughout our proof, we develop several combinatorial criteria of independent interest for detecting various geometric objects on a half-translation surface.