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

Lê, Thang; Paprocki, Jonathan

doi:10.2140/agt.2019.19.3453

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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

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.

Keywords

Kauffman bracket skein module, Chebyshev homomorphism

Mathematical Subject Classification 2010

Primary: 57M25, 57N10

References

Publication

Received: 29 May 2018

Revised: 9 November 2018

Accepted: 28 November 2018

Published: 17 December 2019

Authors

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

Volume 19, issue 7 (2019)