Miraculous cancellations and the quantum Frobenius for $SL_3$ skein modules

Abstract

We construct a quantum Frobenius map for the $SL_3$ 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 $SL_3$ skein algebras of punctured surfaces, using the Frobenius map of Parshall-Wang for the quantum group $\mathcal{O}_q(SL_3).$