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On Kauffman bracket skein modules of marked 3-manifolds and the Chebyshev-Frobenius homomorphism. (English) Zbl 1440.57019

Summary: We study the skein algebras of marked surfaces and the skein modules of marked 3-manifolds. G. Muller [Quantum Topol. 7, No. 3, 435–503 (2016; Zbl 1375.13038)] 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 F. Bonahon and H. Wong [Invent. Math. 204, No. 1, 195–243 (2016; Zbl 1383.57015)] 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.

MSC:

57K31 Invariants of 3-manifolds (including skein modules, character varieties)
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