Multiplicative structure of Kauffman bracket skein module quantizations
Authors:
Doug Bullock and Józef H. Przytycki
Journal:
Proc. Amer. Math. Soc. 128 (2000), 923931
MSC (1991):
Primary 57M99
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 )). For a torus without boundary we obtain a quantization of ``the symmetric homologies" of a torus (equivalently, the coordinate ring of the character variety of ). Presentations are also given for the fourpunctured sphere and twicepunctured torus. We conclude with an investigation of central elements and zero divisors.
 1.
D. Bullock, A finite set of generators for the Kauffman bracket skein algebra, Math. Z., to appear.
 2.
D. Bullock, Rings of characters and the Kauffman bracket skein module, Comm. Math. Helv. 72 (1997), 521542. CMP 98:07
 3.
Robert
D. Horowitz, Characters of free groups represented in the
twodimensional special linear group, Comm. Pure Appl. Math.
25 (1972), 635–649. MR 0314993
(47 #3542)
 4.
Jim
Hoste and Józef
H. Przytycki, A survey of skein modules of 3manifolds, Knots
90 (Osaka, 1990) de Gruyter, Berlin, 1992, pp. 363–379. MR 1177433
(93m:57018)
 5.
A.
V. Odesskiĭ, An analogue of the Sklyanin algebra,
Funktsional. Anal. i Prilozhen. 20 (1986), no. 2,
78–79 (Russian). MR 847152
(87j:17022)
 6.
Józef
H. Przytycki, Skein modules of 3manifolds, Bull. Polish Acad.
Sci. Math. 39 (1991), no. 12, 91–100. MR 1194712
(94g:57011)
 7.
J. H. Przytycki, Introduction to algebraic topology based on knots, Proceedings of Knots 96, (S. Suzuki, ed.) World Scientific (1997) 279297.
 8.
J. H. Przytycki and A. Sikora, On skein algebras and character varieties, eprint: qalg/9705011.
 9.
J. H. Przytycki and A. Sikora, Skein algebra of a group, Proc. Banach Center MiniSemseter on Knot Theory, to appear.
 10.
C.
K. Zachos, Quantum deformations, Quantum groups (Argonne, IL,
1990) World Sci. Publ., Teaneck, NJ, 1991, pp. 62–71. MR 1109753
(92b:17023)
 1.
 D. Bullock, A finite set of generators for the Kauffman bracket skein algebra, Math. Z., to appear.
 2.
 D. Bullock, Rings of characters and the Kauffman bracket skein module, Comm. Math. Helv. 72 (1997), 521542. CMP 98:07
 3.
 R. Horowiz, Characters of free groups represented in the two dimensional linear group, Comm. Pure Appl. Math. 25 (1972) 635649. MR 47:3542
 4.
 J. Hoste and J. H. Przytycki, A survey of skein modules of 3manifolds, Knots 90, de Gruyter (1992) 363379. MR 93m:57018
 5.
 A. Odesskii, An analogue of the Sklyanin algebra, Funct. Anal. Appl. 20 (1986) 7879. MR 87j:17022
 6.
 J. H. Przytycki, Skein modules of 3manifolds, Bull. Polish Acad. Science 39(12) (1991) 91100. MR 94g:57011
 7.
 J. H. Przytycki, Introduction to algebraic topology based on knots, Proceedings of Knots 96, (S. Suzuki, ed.) World Scientific (1997) 279297.
 8.
 J. H. Przytycki and A. Sikora, On skein algebras and character varieties, eprint: qalg/9705011.
 9.
 J. H. Przytycki and A. Sikora, Skein algebra of a group, Proc. Banach Center MiniSemseter on Knot Theory, to appear.
 10.
 C. K. Zachos, Quantum deformations, Proceedings of the Argonne Workshop on Quantum Groups, (T. Curtright, D. Fairle and C. Zachos, eds.) World Scientific (1990). MR 92b:17023
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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:
http://dx.doi.org/10.1090/S0002993999050431
PII:
S 00029939(99)050431
Keywords:
Knot,
link,
3manifold,
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 NSFDMS Postdoctoral Fellowship.
Communicated by:
Ronald A. Fintushel
Article copyright:
© Copyright 1999 American Mathematical Society
