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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
Lê, Thang 1 ; Paprocki, Jonathan 1
@article{10_2140_agt_2019_19_3453,
author = {L\^e, Thang and Paprocki, Jonathan},
title = {On {Kauffman} bracket skein modules of marked 3{\textendash}manifolds and the {Chebyshev{\textendash}Frobenius} homomorphism},
journal = {Algebraic and Geometric Topology},
pages = {3453--3509},
publisher = {mathdoc},
volume = {19},
number = {7},
year = {2019},
doi = {10.2140/agt.2019.19.3453},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.3453/}
}
TY - JOUR AU - Lê, Thang AU - Paprocki, Jonathan TI - On Kauffman bracket skein modules of marked 3–manifolds and the Chebyshev–Frobenius homomorphism JO - Algebraic and Geometric Topology PY - 2019 SP - 3453 EP - 3509 VL - 19 IS - 7 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.3453/ DO - 10.2140/agt.2019.19.3453 ID - 10_2140_agt_2019_19_3453 ER -
%0 Journal Article %A Lê, Thang %A Paprocki, Jonathan %T On Kauffman bracket skein modules of marked 3–manifolds and the Chebyshev–Frobenius homomorphism %J Algebraic and Geometric Topology %D 2019 %P 3453-3509 %V 19 %N 7 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.3453/ %R 10.2140/agt.2019.19.3453 %F 10_2140_agt_2019_19_3453
Lê, Thang; Paprocki, Jonathan. On Kauffman bracket skein modules of marked 3–manifolds and the Chebyshev–Frobenius homomorphism. Algebraic and Geometric Topology, Tome 19 (2019) no. 7, pp. 3453-3509. doi: 10.2140/agt.2019.19.3453
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