Noncommutative trigonometry and the A-polynomial of the trefoil knot

Abstract

The paper shows the computation of the noncommutative generalization of the A-polynomial of the trefoil knot. The classical A-polynomial was introduced by Cooper, Culler, Gillet, Long and Shalen, and was generalized to the context of Kauffman bracket skein modules by the author in joint work with Frohman and Lofaro. A major step in determining the noncommutative version of the A-polynomial of the trefoil is the description of the action of the Kauffman bracket skein algebra of the torus on the skein module of the knot complement. As such, the computation reduces to operations with noncommutative trigonometric functions.