  By generalizing the Kuperberg sl(3) bracket, we construct a graph-valued
analogue of the Homflypt sl(3) invariant for virtual knots. The restriction of
this invariant for classical knots coincides with the usual Homflypt sl(3)
invariant, and for virtual knots and graphs it provides new information that
allows one to prove minimality theorems and to construct new invariants for
free knots. We formulate this new invariant for virtual braids as well, and
show that it leads to the construction of a trace function on the virtual Hecke
algebra. Finally, we show that the Penrose coloring bracket is a special case
of the Kuperberg bracket, and we raise new questions about the extension of the
present work.
