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

TL;DR

This paper constructs a quantum Frobenius map for $SL_3$ skein modules of 3-manifolds at roots of unity, extending previous work on skein algebras and quantum groups with novel threading techniques.

Contribution

It introduces a higher rank quantum Frobenius map for $SL_3$ skein modules, generalizing the Chebyshev-Frobenius homomorphism and building on earlier algebraic constructions.

Findings

Constructed a quantum Frobenius map for $SL_3$ skein modules at roots of unity.

Described the map via threading polynomials along links.

Extended the Frobenius map from skein algebras of punctured surfaces to 3-manifolds.

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 $\mathcal{O}_q(SL_3).$