#### Volume 19, issue 7 (2019)

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

### Thang T Q Lê and Jonathan Paprocki

Algebraic & Geometric Topology 19 (2019) 3453–3509
##### Abstract

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.

##### Keywords
Kauffman bracket skein module, Chebyshev homomorphism
##### Mathematical Subject Classification 2010
Primary: 57M25, 57N10