  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.

Geometric Topology;Quantum Algebra