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 ℱ = A, B | [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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm184-0-2,
author = {Stephen Budden and Roger Fenn},
title = {The equation [B,(A-1)(A,B)] = 0 and virtual knots and links},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {19-29},
zbl = {1070.57003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm184-0-2}
}
Stephen Budden; Roger Fenn. The equation [B,(A-1)(A,B)] = 0 and virtual knots and links. Fundamenta Mathematicae, Tome 184 (2004) pp. 19-29. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm184-0-2/