The Apolynomial from the noncommutative viewpoint
Authors:
Charles Frohman, Razvan Gelca and Walter LoFaro
Journal:
Trans. Amer. Math. Soc. 354 (2002), 735747
MSC (1991):
Primary 57M25, 58B30, 46L87
Published electronically:
October 3, 2001
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Abstract: The paper introduces a noncommutative generalization of the Apolynomial of a knot. This is done using the Kauffman bracket skein module of the knot complement, and is based on the relationship between skein modules and character varieties. The construction is possible because the Kauffman bracket skein algebra of the cylinder over the torus is a subalgebra of the noncommutative torus. The generalized version of the Apolynomial, called the noncommutative Aideal, consists of a finitely generated ideal of polynomials in the quantum plane. Some properties of the noncommutative Aideal and its relationships with the Apolynomial and the Jones polynomial are discussed. The paper concludes with the description of the examples of the unknot, and the right and lefthanded trefoil knots.
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Additional Information
Charles Frohman
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
Email:
frohman@math.uiowa.edu
Razvan Gelca
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409 and Institute of Mathematics of The Romanian Academy, Bucharest, Romania
Email:
rgelca@math.ttu.edu
Walter LoFaro
Affiliation:
Department of Mathematics and Computing, University of WisconsinStevens Point, Stevens Point, Wisconsin 54481
Email:
Walter.LoFaro@uwsp.edu
DOI:
http://dx.doi.org/10.1090/S0002994701028896
PII:
S 00029947(01)028896
Keywords:
Kauffman bracket,
skein modules,
Apolynomial,
character varieties,
noncommutative geometry
Received by editor(s):
March 14, 2001
Received by editor(s) in revised form:
May 7, 2001
Published electronically:
October 3, 2001
Article copyright:
© Copyright 2001
American Mathematical Society
