Miraculous cancellations and the quantum Frobenius for SL₃ skein modules

We construct a quantum Frobenius map for the SL₃ skein module of any oriented 3-manifold specialized at a root of unity, and describe the map by way of threading certain polynomials along links. The homomorphism is a higher rank version of the Chebyshev-Frobenius homomorphism of Bonahon-Wong. The strategy builds on a previous construction of the Frobenius map for SL₃ skein algebras of punctured surfaces, using the Frobenius map of Parshall-Wang for the quantum group Oq (SL₃).