The A-polynomial from the noncommutative viewpoint

Charles Frohman, Rǎzvan Gelca, Walter Lofaro

Research output: Contribution to journalArticle

33 Scopus citations

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.

Original languageEnglish
Pages (from-to)735-747
Number of pages13
JournalTransactions of the American Mathematical Society
Volume354
Issue number2
DOIs
StatePublished - 2002

Keywords

  • A-polynomial
  • Character varieties
  • Kauffman bracket
  • Noncommutative geometry
  • Skein modules

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