The Chebyshev-Frobenius homomorphism for stated skein modules of 3-manifolds

TL;DR

This paper introduces a Chebyshev-Frobenius homomorphism for stated skein modules of 3-manifolds, extending previous algebraic constructions and establishing its uniqueness and compatibility with splitting homomorphisms.

Contribution

It generalizes the Chebyshev-Frobenius homomorphism to 3-manifolds and proves its uniqueness and compatibility with splitting homomorphisms, linking skein modules to quantum group structures.

Findings

Existence of a Chebyshev-Frobenius homomorphism for 3-manifolds' skein modules.

The homomorphism commutes with splitting homomorphisms.

Uniqueness of the homomorphism as an extension of the dual Frobenius map.

Abstract

We study the stated skein modules of marked 3-manifolds. We generalize the splitting homomorphism for stated skein algebras of surfaces to a splitting homomorphism for stated skein modules of 3-manifolds. We show that there exists a Chebyshev-Frobenius homomorphism for the stated skein modules of 3-manifolds which extends the Chebyshev homomorphism of the skein algebras of unmarked surfaces originally constructed by Bonahon and Wong. Additionally, we show that the Chebyshev-Frobenius map commutes with the splitting homomorphism. This is then used to show that in the case of the stated skein algebra of a surface, the Chebyshev-Frobenius map is the unique extension of the dual Frobenius map (in the sense of Lusztig) of $\mathcal{O}_{q^2}(SL(2))$ through the triangular decomposition afforded by an ideal triangulation of the surface. In particular, this gives a skein theoretic construction of the Hopf dual of Lusztig's Frobenius homomorphism. A second conceptual framework is given, which shows that the Chebyshev-Frobenius homomorphism for the stated skein algebra of a surface is the unique restriction of the Frobenius homomorphism of quantum tori through the quantum trace map.