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Transactions of the American Mathematical Society

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Skein algebras of surfaces


Authors: Józef H. Przytycki and Adam S. Sikora
Journal: Trans. Amer. Math. Soc. 371 (2019), 1309-1332
MSC (2010): Primary 57M25, 57M27
DOI: https://doi.org/10.1090/tran/7298
Published electronically: August 21, 2018
MathSciNet review: 3885180
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Abstract: We show that the Kauffman bracket skein algebra of any oriented surface $ F$ (possibly with marked points in its boundary) has no zero divisors and that its center is generated by knots parallel to the unmarked components of the boundary of $ F$. Furthermore, we show that skein algebras are Noetherian and Ore. Our proofs rely on certain filtrations of skein algebras induced by pants decompositions of surfaces. We prove some basic algebraic properties of the associated graded algebras along the way.


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Additional Information

Józef H. Przytycki
Affiliation: Department of Mathematics, George Washington University, Washington, DC 20052 — and — Department of Mathematics, Physics and Informatics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
Email: przytyck@gwu.edu

Adam S. Sikora
Affiliation: Department of Mathematics, University at Buffalo, SUNY, Buffalo, New York 14260
Email: asikora@buffalo.edu

DOI: https://doi.org/10.1090/tran/7298
Keywords: Kauffman bracket skein module, skein algebra, Dehn--Thurston numbers
Received by editor(s): May 9, 2016
Received by editor(s) in revised form: January 25, 2017, January 26, 2017, and May 25, 2017
Published electronically: August 21, 2018
Additional Notes: The first author acknowledges support of the Simons Foundation Collaboration Grant for Mathematicians 316446.
The second author acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).
Article copyright: © Copyright 2018 American Mathematical Society