Non-commutative trigonometry and the A-polynomial of the trefoil knot

Abstract

The non-commutative generalization of the A-polynomial of a knot of Cooper, Culler, Gillet, Long and Shalen [ 4 ] was introduced in [ 6 ]. This generalization consists of a finitely generated left ideal of polynomials in the quantum plane, the non- commutative A-ideal, and was defined based on Kauffman bracket skein modules, by deforming the ideal generated by the A-polynomial with respect to a parameter. The deformation was possible because of the relationship between the skein module with the variable t of the Kauffman bracket evaluated at −1 and the SL (2, C )-character variety of the fundamental group, which was explained in [ 2 ]. The purpose of the present paper is to compute the non-commutative A-ideal for the left- and right- handed trefoil knots. As will be seen below, this reduces to trigonometric operations in the non-commutative torus, the main device used being the product-to-sum formula for non-commutative cosines.