On Kauffman bracket skein modules of marked 3-manifolds and the Chebyshev-Frobenius homomorphism

TL;DR

This paper extends the understanding of skein modules and algebras of marked 3-manifolds, introducing a Chebyshev-Frobenius homomorphism and analyzing its properties, including transparency and the algebra's center.

Contribution

It generalizes Muller's embedding result to unmarked boundary components and develops a surgery theory to extend the Chebyshev homomorphism to skein modules.

Findings

The Chebyshev-Frobenius homomorphism's image is transparent or skew-transparent.

The center of the skein algebra is computed when the quantum parameter is not a root of unity.

Extension of Muller's embedding to surfaces with unmarked boundary components.

Abstract

In this paper we study the skein algebras of marked surfaces and the skein modules of marked 3-manifolds. Muller showed that skein algebras of totally marked surfaces may be embedded in easy to study algebras known as quantum tori. We first extend Muller's result to permit marked surfaces with unmarked boundary components. The addition of unmarked components allows us to develop a surgery theory which enables us to extend the Chebyshev homomorphism of Bonahon and Wong between skein algebras of unmarked surfaces to a "Chebyshev-Frobenius homomorphism" between skein modules of marked 3-manifolds. We show that the image of the Chebyshev-Frobenius homomorphism is either transparent or skew-transparent. In addition, we make use of the Muller algebra method to calculate the center of the skein algebra of a marked surface when the quantum parameter is not a root of unity.