The A-polynomial from the noncommutative viewpoint

Authors:
Charles Frohman, Razvan Gelca and Walter LoFaro

Journal:
Trans. Amer. Math. Soc. **354** (2002), 735-747

MSC (1991):
Primary 57M25, 58B30, 46L87

Published electronically:
October 3, 2001

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Abstract: The paper introduces a noncommutative generalization of the A-polynomial 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 A-polynomial, called the noncommutative A-ideal, consists of a finitely generated ideal of polynomials in the quantum plane. Some properties of the noncommutative A-ideal and its relationships with the A-polynomial and the Jones polynomial are discussed. The paper concludes with the description of the examples of the unknot, and the right- and left-handed 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 Wisconsin-Stevens Point, Stevens Point, Wisconsin 54481

Email:
Walter.LoFaro@uwsp.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02889-6

Keywords:
Kauffman bracket,
skein modules,
A-polynomial,
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