# Orthodiagonal anti-involutive Kokotsakis polyhedra

## TL;DR

This paper proves Stachel's conjecture regarding the reducibility of a specific resultant in orthodiagonal anti-involutive Kokotsakis polyhedra, confirming their flexibility and providing explicit parameterizations.

## Contribution

It demonstrates the reducibility of the resultant for these polyhedra, confirming their flexibility and offering explicit parameterizations in elementary and elliptic functions.

## Key findings

- The resultant associated with the polyhedron is reducible.
- Such polyhedra are flexible and have non-empty parameter space.
- Explicit parameterizations of flexion are provided.

## Abstract

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.