Weyl Algebras and Knots

Abstract

In this paper we push forward results on the invariant $\mathcal{F}$-module of a virtual knot investigated by the first named author where $\mathcal{F}$ is the algebra with two invertible generators $A,B$ and one relation $A^{-1}{B}^{-1}AB-B^{-1}AB=B{A}^{-1}{B}^{-1}A-A$. 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.

18 pages, 5 figures, accepted by Journal of Geometry and Physics

References in corpus (2)

• On Alexander-Conway Polynomials for Virtual Knots and Links
• A Polynomial Invariant for Flat Virtual Links

Cited by in corpus (8)

• Cobordism of knots on surfaces
• Link invariants from finite racks
• On tabulating virtual strings
• New Invariants of Long Virtual Knots
• An Extended Bracket Polynomial for Virtual Knots and Links
• On the q-meromorphic Weyl algebra
• Why FHilb is Not an Interesting (Co)Differential Category
• Coverings, composites and cables of virtual strings