Rigidity of the saddle connection complex

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