The A-polynomial from the noncommutative viewpoint
The A-polynomial from the noncommutative viewpoint
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非交换观点的 A 多项式
DOI：
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
Walter Lofaro
中科院分区：
文献类型：
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作者：
C. Frohman;R. Gelca;Walter Lofaro
本文给出了纽结的A-多项式的一个非交换推广。这是使用结补的考夫曼括号绞链模块完成的，并且是基于绞链模块和字符变体之间的关系。这种构造是可能的，因为环面上圆柱的Kauffman括号skein代数是非交换环面的子代数。A-多项式的广义形式，称为非对易A-理想，由量子平面中多项式的一个非对易生成理想组成。讨论了非交换A-理想的一些性质及其与A-多项式和Jones多项式的关系。本文最后描述的例子的unknot，和右手和左手三叶结。
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.