Generalized Quaternions and Invariants of Virtual Knots and Links

In this paper we will call this the fundamental algebra and the single relation will be called the fundamental relation or equation. This relation arises naturally from attempts to find representations of the braid group. Representations of the fundamental algebra as matrices can be used to define representations of the virtual braid group and invariants of virtual knots and links. In [BuF] we found a complete set of conditions for two classic quaternions, A,B to be solutions of the fundamental equation. In this paper this result is generalised to give necessary and sufficient conditions for generalized quaternions to satisfy the fundamental relation, except in the case of all 2×2 matrices where only sufficient conditions are given. Particularly, we define two 4-variable polynomials of virtual knots and links. In addition, we give conclusive proof of the fact, only hinted at in earlier papers, that invariants defined in this manner do not give any new invariants for classical knots and links.