For a half-translation surface $(S,q)$(S,q)$(S,q)$, the associated saddle connection complex $\mathcal{A}(S,q)$A(S,q)$\mathcal{A}(S,q)$ is the simplicial complex where vertices are the saddle connections on $(S,q)$(S,q)$(S,q)$, 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 :\mathcal{A}(S,q)\to \mathcal{A}(S^{\prime },q^{\prime })$$\varphi :\mathcal{A}(S,q)\to \mathcal{A}(S^{\prime },q^{\prime })$ between saddle connection complexes is induced by an affine diffeomorphism $F:(S,q)\to (S^{\prime },q^{\prime })$$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.

中文翻译：

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鞍形连接复合体的刚性

对于半平移曲面$(\mathrm{小号},q)$（小号, q)$(S,q)$，相关的鞍连接复合体 $\mathcal{一个}(\mathrm{小号},q)$一个（小号, q)$\mathcal{A}(S,q)$是单纯复形，其中顶点是上的鞍连接 $(\mathrm{小号},q)$（小号, q)$(S,q)$，由成对不相交的鞍连接集跨越的单纯形。这个复合体可以自然地被认为是弧复合体的诱导子复合体。我们证明任何单纯同构$\phi ：\mathcal{一个}(\mathrm{小号},q)\to \mathcal{一个}({\mathrm{小号}}^{\text{'}},q^{\text{'}})$$\varphi :\mathcal{A}(S,q)\to \mathcal{A}(S^{\prime },q^{\prime })$鞍连接复合体之间是由仿射微分同胚引起的$F：(\mathrm{小号},q)\to ({\mathrm{小号}}^{\text{'}},q^{\text{'}})$$F:(S,q)\to (S^{\prime },q^{\prime })$. 特别是，这表明鞍连接复形是半平移曲面的仿射等价类的完全不变量。在整个证明过程中，我们开发了几个独立感兴趣的组合标准，用于检测半平移表面上的各种几何对象。