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2019 On Kauffman bracket skein modules of marked $3$–manifolds and the Chebyshev–Frobenius homomorphism
Thang T Q Lê, Jonathan Paprocki
Algebr. Geom. Topol. 19(7): 3453-3509 (2019). DOI: 10.2140/agt.2019.19.3453

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.

Citation

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Thang T Q Lê. Jonathan Paprocki. "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

Information

Received: 29 May 2018; Revised: 9 November 2018; Accepted: 28 November 2018; Published: 2019
First available in Project Euclid: 3 January 2020

zbMATH: 07162212
MathSciNet: MR4045358
Digital Object Identifier: 10.2140/agt.2019.19.3453

Subjects:
Primary: 57M25, 57N10

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.19 • No. 7 • 2019
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