We study the properties of Kokotsakis polyhedra of orthodiagonal anti-involutive type. Stachel conjectured that a certain resultant connected to a polynomial system describing flexion of a Kokotsakis polyhedron must be reducible. Izmestiev \cite{izmestiev2016classification} showed that a polyhedron of the orthodiagonal anti-involutive type is the only possible candidate to disprove Stachel's conjecture. We show that the corresponding resultant is reducible, thereby confirming the conjecture. We do it in two ways: by factorization of the corresponding resultant and providing a simple geometric proof. We describe the space of parameters for which such a polyhedron exists and show that this space is non-empty. We show that a Kokotsakis polyhedron of orthodiagonal anti-involutive type is flexible and give explicit parameterizations in elementary functions and in elliptic functions of its flexion.

中文翻译：

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正对角反对合 Kokotsakis 多面体

我们研究了正对角反对合型 Kokotsakis 多面体的性质。Stachel 推测，连接到描述 Kokotsakis 多面体弯曲的多项式系统的某个结果一定是可约的。Izmestiev \cite{izmestiev2016classification} 表明正对角反对合类型的多面体是反驳 Stachel 猜想的唯一可能候选者。我们证明相应的结果是可约的，从而证实了猜想。我们通过两种方式做到这一点：通过对相应结果进行因式分解并提供简单的几何证明。我们描述了存在这样一个多面体的参数空间，并表明这个空间是非空的。