Weyl Algebras and Knots

Abstract

In this paper we push forward results on the invariant ${\cal F}$-module of a virtual knot investigated by the first named author where ${\cal F}$ is the algebra with two invertible generators $A,B$ and one relation $A^{-1}B^{-1}AB-B^{-1}AB= BA^{-1}B^{-1}A-A$. For flat knots and links the two sides of the relation equation are put equal to unity and the algebra becomes the Weyl algebra. If this is perturbed and the two sides of the relation equation are put equal to a general element, $q$, of the ground ring, then the resulting module lays claim to be the correct generalization of the Alexander module. Many finite dimensional representations are given together with calculations.