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

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.

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Kauffman bracket skein module, Chebyshev homomorphism
Mathematical Subject Classification 2010
Primary: 57M25, 57N10
Received: 29 May 2018
Revised: 9 November 2018
Accepted: 28 November 2018
Published: 17 December 2019
Thang T Q Lê
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States
Jonathan Paprocki
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States