Weyl Algebras and Knots

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.Comment: 18 pages, 5 figures, accepted by Journal of Geometry and Physic

James bundles

We study cubical sets without degeneracies, which we call {square}-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a {square}-set C has an infinite family of associated {square}-sets Ji(C), for i = 1, 2, ..., which we call James complexes. There are mock bundle projections pi: |Ji(C)| -> |C| (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of {Omega}(S2). The algebra of these classes mimics the algebra of the cohomotopy of {Omega}(S2) and the reduction to cohomology defines a sequence of natural characteristic classes for a {square}-set. An associated map to BO leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation

Virtual Knot Theory --Unsolved Problems

This paper is an introduction to the theory of virtual knots and links and it gives a list of unsolved problems in this subject.Comment: 33 pages, 7 figures, LaTeX documen