The equation $[B,(A-1)(A,B)]=0$ and virtual knots and links
Volume 184 / 2004
Fundamenta Mathematicae 184 (2004), 19-29
MSC: 57M25, 57M27.
DOI: 10.4064/fm184-0-2
Abstract
Let $A$, $B$ be invertible, non-commuting elements of a ring $R$. Suppose that $A-1$ is also invertible and that the equation $[B,(A-1)(A,B)]=0$ called the fundamental equation is satisfied. Then this defines a representation of the algebra ${\mathcal F}=\{ A, B\mid [B,(A-1)(A,B)]=0\} $. An invariant $R$-module can then be defined for any diagram of a (virtual) knot or link. This halves the number of previously known relations and allows us to give a complete solution in the case when $R$ is the quaternions.
