We study the skein algebras of marked surfaces and the skein modules of marked –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 –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.
"On Kauffman bracket skein modules of marked $3$–manifolds and the Chebyshev–Frobenius homomorphism." Algebr. Geom. Topol. 19 (7) 3453 - 3509, 2019. https://doi.org/10.2140/agt.2019.19.3453