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Multiplicative structure of Kauffman bracket skein module quantizations


Authors: Doug Bullock and Józef H. Przytycki
Journal: Proc. Amer. Math. Soc. 128 (2000), 923-931
MSC (1991): Primary 57M99
DOI: https://doi.org/10.1090/S0002-9939-99-05043-1
Published electronically: July 28, 1999
MathSciNet review: 1625701
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Abstract: We describe, for a few small examples, the Kauffman bracket skein algebra of a surface crossed with an interval. If the surface is a punctured torus the result is a quantization of the symmetric algebra in three variables (and an algebra closely related to a cyclic quantization of $U(\mathfrak{so}_3$)). For a torus without boundary we obtain a quantization of ``the symmetric homologies" of a torus (equivalently, the coordinate ring of the $SL_2(\mathbb{C})$-character variety of $\mathbb{Z}\oplus\mathbb{Z}$). Presentations are also given for the four-punctured sphere and twice-punctured torus. We conclude with an investigation of central elements and zero divisors.


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Additional Information

Doug Bullock
Affiliation: Department of Mathematics, The George Washington University, Washington, DC 20052
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: bullock@math.umd.edu

Józef H. Przytycki
Affiliation: Department of Mathematics, The George Washington University, Washington, DC 20052
Email: przytyck@math.gwu.edu

DOI: https://doi.org/10.1090/S0002-9939-99-05043-1
Keywords: Knot, link, 3-manifold, skein module
Received by editor(s): November 17, 1997
Received by editor(s) in revised form: May 5, 1998
Published electronically: July 28, 1999
Additional Notes: The first author is supported by an NSF-DMS Postdoctoral Fellowship.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 1999 American Mathematical Society

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